Recent zbMATH articles in MSC 42 https://zbmath.org/atom/cc/42 2021-11-25T18:46:10.358925Z Unknown author Werkzeug A lower bound for the discriminant of polynomials related to Chebyshev polynomials https://zbmath.org/1472.05158 2021-11-25T18:46:10.358925Z "Filipovski, Slobodan" https://zbmath.org/authors/?q=ai:filipovski.slobodan The discriminant of a polynomial $$f=a\prod_{i=0}^m (x-\alpha_i)$$ is defined by $D(f)=a^{2m-2}\prod_{1\leq i<j\leq m} (\alpha_j-\alpha_i)^2.$ The discriminant is a useful tool to get information about the roots of a high-degree polynomial. In this paper, the author gives a lower bound for the discriminant of the polynomials $$\{G_{k,i}\}$$ defined by $$G_{k,0}(x)=1$$, $$G_{k,1}(x)=x+1$$ and then recursively $G_{k,i+2}(x)=xG_{k,i+1}(x)-(k-1)G_{k,i}(x).$ These polynomials have been used for the study of graphs and their girth (this is the length of the shortest circuit in the graph), specially due to its relation with the Moore Bound $M_{d,k}=\begin{cases} 1+d\frac{(d-1)^{k-1}-1}{d-2}& d>2\\ 2k+1& d=2 \end{cases},$ which relates the degree, the order, the diameter and the girth of a graph. The discriminant of these polynomials have been used to prove the existence of graphs with certain degree or girth as in [\textit{C. Delorme} and \textit{G. Pineda-Villavicencio}, Electron. J. Comb. 17, No. 1, Research Paper R143, 25 p. (2010; Zbl 1204.05043)]. While all of this is addressed in the second section of the paper and the author suggests a path to proof his main result in a graph-theoretical way, most of the document is devoted to proof the result in a more elementary way. The main idea is to use orthogonality of some polynomials related to $$\{G_{k,i}\}$$ and from there stablish a relation through simple inequalities with the discriminant of $$G_{k,d}$$. While the notation could be a little heavy, the results are quite simple to follow, leading to the following bound for the discriminant: $D(G_{k,d})>d^d(k-2)\left[\sqrt{k(k-1)^2-2}\right]^{d-2}.$ Knit product of finite groups and sampling https://zbmath.org/1472.20007 2021-11-25T18:46:10.358925Z "García, Antonio G." https://zbmath.org/authors/?q=ai:garcia.antonio-g "Hernández-Medina, Miguel A." https://zbmath.org/authors/?q=ai:hernandez-medina.miguel-angel "Ibort, Alberto" https://zbmath.org/authors/?q=ai:ibort.alberto Summary: A finite sampling theory associated with a unitary representation of a finite non-abelian group $${\mathbf{G}}$$ on a Hilbert space is established. The non-abelian group $${\mathbf{G}}$$ is a knit product $${\mathbf{N}}\bowtie{\mathbf{H}}$$ of two finite subgroups $${\mathbf{N}}$$ and $${\mathbf{H}}$$ where at least $${\mathbf{N}}$$ or $${\mathbf{H}}$$ is abelian. Sampling formulas where the samples are indexed by either $${\mathbf{N}}$$ or $${\mathbf{H}}$$ are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space $$\ell^2({\mathbf{G}})$$ having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results. On approximation of fractional derivatives of functions by trigonometric polynomials in $$L_p$$, $$0<p<1$$ https://zbmath.org/1472.26001 2021-11-25T18:46:10.358925Z "Kolomoitsev, Yu. S." https://zbmath.org/authors/?q=ai:kolomoitsev.yurii-s "Lomako, T. V." https://zbmath.org/authors/?q=ai:lomako.tetiana Summary: In this paper we obtain a generalization to the case of fractional derivatives of known results on the approximation of the integer order derivatives for functions in the spaces $$L_p$$, $$0 < p < 1$$. Nonspectrality of certain self-affine measures on $$\mathbb{R}^3$$ https://zbmath.org/1472.28003 2021-11-25T18:46:10.358925Z "Gao, Gui-Bao" https://zbmath.org/authors/?q=ai:gao.guibao Summary: We will determine the nonspectrality of self-affine measure $$\mu_{B, D}$$ corresponding to $$B = \operatorname{diag} [p_1, p_2, p_3]$$ ($$p_1 \in(2 \mathbb Z + 1) \smallsetminus \{\pm 1 \}$$, $$p_2 \in 2 \mathbb Z \smallsetminus \{0 \}$$), and $$D = \{0, e_1, e_2, e_3 \}$$ in the space $$\mathbb{R}^3$$ is supported on $$T(B, D)$$, where $$e_1, e_2$$, and $$e_3$$ are the standard basis of unit column vectors in $$\mathbb{R}^3$$, and there exist at most 4 mutually orthogonal exponential functions in $$L^2(\mu_{B, D})$$, where the number 4 is the best. This generalizes the known results on the spectrality of self-affine measures. Spectrality of generalized Sierpinski-type self-affine measures https://zbmath.org/1472.28007 2021-11-25T18:46:10.358925Z "Liu, Jing-Cheng" https://zbmath.org/authors/?q=ai:liu.jingcheng "Zhang, Ying" https://zbmath.org/authors/?q=ai:zhang.ying.3|zhang.ying.4|zhang.ying|zhang.ying.1|zhang.ying.5|zhang.ying.2 "Wang, Zhi-Yong" https://zbmath.org/authors/?q=ai:wang.zhiyong.2|wang.zhiyong.1 "Chen, Ming-Liang" https://zbmath.org/authors/?q=ai:chen.ming-liang Summary: In this work, we study the spectral property of generalized Sierpinski-type self-affine measures $$\mu_{M,D}$$ on $$\mathbb{R}^2$$ generated by an expanding integer matrix $$M\in M_2(\mathbb{Z})$$ with $$\det(M)\in 3\mathbb{Z}$$ and a non-collinear integer digit set $$D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}$$ with $$\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb{Z}$$. We give the sufficient and necessary conditions for $$\mu_{M,D}$$ to be a spectral measure, i.e., there exists a countable subset $$\Lambda\subset \mathbb{R}^2$$ such that $$E(\Lambda)=\{e^{2\pi i\langle\lambda,x \rangle}:\lambda\in\Lambda\}$$ forms an orthonormal basis for $$L^2(\mu_{M,D})$$. This completely settles the spectrality of the self-affine measure $$\mu_{M,D}$$. Spectrality of a class of self-affine measures and related digit sets https://zbmath.org/1472.28013 2021-11-25T18:46:10.358925Z "Yang, Ming-Shu" https://zbmath.org/authors/?q=ai:yang.ming-shu Summary: This work investigates the spectrality of a self-affine measure $$\mu_{M,D}$$ and the related digit set $$D$$ in the case when $$|\mathrm{det}(M)|=p^{\alpha}$$ is a prime power and $$|D|=p$$ is a prime, where $$\alpha\in{\mathbb{N}}$$, and $$\mu_{M,D}$$ is generated by an expanding matrix $$M\in M_n({\mathbb{Z}})$$ and a digit set $$D\subset\mathbb{Z}^n$$ of cardinality |$$D$$|. We obtain that $$\mu_{M,D}$$ is a spectral measure and $$D$$ is a spectral set if one nonzero element in $$D$$ satisfies certain mild conditions. This is based on the property of vanishing sums of roots of unity and a residue system in number theory. The result here extends the corresponding known results and provides some supportive evidence for a conjecture of Dutkay, Han, and Jorgensen. Asymptotics of Chebyshev polynomials. IV: Comments on the complex case https://zbmath.org/1472.30001 2021-11-25T18:46:10.358925Z "Christiansen, Jacob S." https://zbmath.org/authors/?q=ai:christiansen.jacob-stordal "Simon, Barry" https://zbmath.org/authors/?q=ai:simon.barry.1 "Zinchenko, Maxim" https://zbmath.org/authors/?q=ai:zinchenko.maxim The Chebyshev polynomial $$T_n$$ of a compact infinite set $$E\subset{\mathbb C}$$ is that monic polynomial of degree-$$n$$ which minimizes $${\|P_n\|}_E$$ over all degree $$n$$ monic polynomials $$P_n$$, where $${\|\cdot\|}_E$$ denotes the supremum norm on $$E$$. In this paper, which is the fourth part of a series of papers (the second joint with Yuditskii), all of them devoted to Chebyshev polynomials and related problems, the authors present some results for rather general sets $$E$$ in the complex plane. On the one hand, they prove some interesting results concerning the asymptotics of the zeros of $$T_n$$, and on the other hand, they give explicit Totik-Widom upper bounds for certain complex sets $$E$$. For Part III, see [the authors, Oper. Theory: Adv. Appl. 276, 231--246 (2020; Zbl 1448.41026)]. A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: bounds, asymptotics and zeros https://zbmath.org/1472.33007 2021-11-25T18:46:10.358925Z "Bracciali, Cleonice F." https://zbmath.org/authors/?q=ai:bracciali.cleonice-f "da Silva, Jéssica V." https://zbmath.org/authors/?q=ai:da-silva.jessica-v "Ranga, A. Sri" https://zbmath.org/authors/?q=ai:ranga.a-sri Authors' abstract: This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with respect to these parameters. Criteria for knowing when the zeros of the (Sobolev) orthogonal polynomials and also the zeros of their derivatives stay within the unit disk have also been addressed. By numerical experiments some further information on the parameters is also found so that the zeros remain within the unit disk. A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities https://zbmath.org/1472.35016 2021-11-25T18:46:10.358925Z "Martínez-Perales, Javier C." https://zbmath.org/authors/?q=ai:martinez-perales.javier-c Summary: The main result of this paper supports a conjecture by Pérez and Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincaré-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by Pérez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range $$p<1$$. As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects. Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting. Fundamental solutions and decay rates for evolution problems on the torus $$\mathbb{T}^n$$ https://zbmath.org/1472.35052 2021-11-25T18:46:10.358925Z "Guiñazú, Alex" https://zbmath.org/authors/?q=ai:guinazu.alex "Vergara, Vicente" https://zbmath.org/authors/?q=ai:vergara.vicente Summary: In this paper we study large-time behavior evolution problems on the n-dimensional torus $$\mathbb{T}^n$$, $$n\geq 1$$. Here we analyze the solutions to these problems, studying their regularity and obtaining estimates of them. The main tools we use is the toroidal Fourier transform, together with Fourier series and a version of the Hardy-Littlewood inequality, applied to our case of the n-dimensional torus $$\mathbb{T}^n$$. We use this inequality to find an estimate of solutions to evolution problems. Global wellposedness and large time behavior of solutions to the $$N$$-dimensional compressible Oldroyd-B model https://zbmath.org/1472.35057 2021-11-25T18:46:10.358925Z "Zhai, Xiaoping" https://zbmath.org/authors/?q=ai:zhai.xiaoping "Li, Yongsheng" https://zbmath.org/authors/?q=ai:li.yongsheng The authors consider the compressible Oldroyd-B model written as $$\partial _{t}a+\operatorname{div} u=-\operatorname{div}(au)$$, $$\partial _{t}\eta -\varepsilon \Delta \eta =-\operatorname{div}(\eta u)$$, $$\partial _{t}\mathbb{T}+\frac{A_{0}}{2\lambda _{1}}\mathbb{T}+(u\cdot \nabla )\mathbb{T}-\varepsilon \Delta \mathbb{T}=\frac{\kappa A_{0}}{ 2\lambda _{1}}\eta \mathbb{I}d+F(\mathbb{T},u)$$, $$\partial _{t}u+u\cdot \nabla u-\mu \Delta u-(\lambda +\mu )\nabla \operatorname{div}u+\nabla a=\operatorname{div}\mathbb{T} -\kappa L\nabla \eta +G(a,u,\eta ,\mathbb{T})$$, with $$F(\mathbb{T} ,u)=(\nabla u\mathbb{T}+\mathbb{T}\nabla u)-\mathbb{T}\operatorname{div}u$$ and $$G(a,u,\eta , \mathbb{T})=k(a)\nabla a-I(a)(\mu u+(\lambda +\mu )\nabla \operatorname{div}u)-I(a)(\operatorname{div}\mathbb{T}-\nabla \eta )-\zeta (1-I(a))\eta \nabla \eta$$; posed in $$\mathbb{R}^{+}\times \mathbb{R}^{n}$$, $$n=2,3$$. Here $$a=1-\rho$$, $$\rho = \rho (t,x)\in \mathbb{R}^{+}$$ is the density function of the fluid, $$u=u(t,x)\in \mathbb{R}^{n}$$ is the velocity. $$\mathbb{T}=(\mathbb{T}_{i,j}(t,x))$$, $$1\leq i,j\leq n$$, is a symmetric matrix function representing the extra stress tensor, $$\eta =\eta (t,x)\in \mathbb{R}^{+}$$ represents the polymer number density defined as the integral of a probability density function $$\psi$$ which is governed by the Fokker-Plank equation, with respect to the conformation vector, $$\mu >0$$ and $$\lambda$$ are the viscosity constants which are supposed to satisfy $$n\lambda +2\mu \geq 0$$, the parameters $$\kappa$$, $$\varepsilon$$, $$A_{0}$$, $$\lambda _{1}$$ are positive numbers, whereas $$\zeta \geq 0$$ and $$L\geq 0$$ with $$\zeta +L\neq 0$$. Initial conditions are added to $$(a,u,\eta ,\mathbb{T})$$. The authors first prove a local existence result for this problem, assuming $$1<p<2n$$ and that for any initial data $$u_{0}\in \overset{.}{B}_{p,1}^{n/p-1}(\mathbb{R}^{n})$$, $$(\eta _{0},\mathbb{T}_{0})\in \overset{.}{B}_{p,1}^{n/p}(\mathbb{R}^{n})$$ and $$a_{0}\in \overset{.}{B}_{p,1}^{n/p}(\mathbb{R}^{n})$$ with $$1+a_{0}$$ bounded away from zero.\ There exists $$T>0$$ such that the above problem has a unique solution with \begin{align*} & a\in C_{b}([0,T];\overset{.}{B}_{p,1}^{n/p}),\; u\in C_{b}([0,T];\overset{.}{B}_{p,1}^{n/p-1})\cap L^{1}([0,T];\overset{.}{B} _{p,1}^{n/p+1}), \\ & (\eta ,\mathbb{T})\in C_{b}([0,T];\overset{.}{B} _{p,1}^{n/p})\cap L^{1}([0,T];\overset{.}{B}_{p,1}^{n/p+2}), \; \mathbb{T}\in L^{1}([0,T];\overset{.}{B}_{p,1}^{n/p}). \end{align*} The authors then prove a global existence result to this problem, assuming further hypotheses on $$p$$ and on the initial data. The last main result of the problem proves optimal decay estimates on this global solution with respect to time. The authors observe that the equations of the polymer number density $$\eta$$ and the extra stress tensor $$\mathbb{T}$$ are two heat-flow type, and that, when the polymer number density $$\eta$$ and the extra stress tensor $$\mathbb{T}$$ vanish, the above problem reduces to the barotropic Navier-Stokes equations. They thus refer to previous results of the literature for the local existence result. For the proof of the second main result, the authors use a continuity argument, establishing a priori bounds considering the low frequency and the high frequency parts of the solution. For the decay estimates, the authors establish a Lyapunov-type inequality in time for energy norms. Modular maximal estimates of Schrödinger equations https://zbmath.org/1472.35070 2021-11-25T18:46:10.358925Z "Ho, Kwok-Pun" https://zbmath.org/authors/?q=ai:ho.kwok-pun Summary: This paper offers the maximal estimates of the solutions of some initial value problems on modular spaces. Our results include the estimates for the solutions of Schrödinger equation. Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentials https://zbmath.org/1472.35257 2021-11-25T18:46:10.358925Z "Blair, Matthew D." https://zbmath.org/authors/?q=ai:blair.matthew-d "Sire, Yannick" https://zbmath.org/authors/?q=ai:sire.yannick "Sogge, Christopher D." https://zbmath.org/authors/?q=ai:sogge.christopher-d Summary: We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, $$H_V=-\Delta_g+V(x)$$, on compact Riemannian manifolds $$(M,g)$$ of dimension $$n\geq 2$$, which extend the results of the third author [J. Funct. Anal. 77, No. 1, 123--138 (1988; Zbl 0641.46011)] corresponding to the case where $$V\equiv 0$$. We are able to handle critically singular potentials and consequently assume that $$V\in L^{\frac{n}{2}}(M)$$ and/or $$V\in\mathcal{K}(M)$$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $$V\equiv 0$$ that go back to the third author [loc. cit.] as well as ones which arose in the work of \textit{C. E. Kenig} et al. [Duke Math. J. 55, 329--347 (1987; Zbl 0644.35012)] in the study of uniform Sobolev estimates'' in $$\mathbb{R}^n$$. We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural $$L^p\rightarrow L^p$$ spectral multiplier theorems under the assumption that $$V\in L^{\frac{n}{2}}(M)\cap\mathcal{K}(M)$$. Moreover, we can also obtain natural analogs of the original \textit{R. S. Strichartz} estimates [Duke Math. J. 44, 705--714 (1977; Zbl 0372.35001)] for solutions of $$(\partial_t^2-\Delta +V)u=0$$. We also are able to obtain analogous results in $$\mathbb{R}^n$$ and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in $$\mathbb{R}^n$$ (e.g., \textit{A. D. Ionescu} and \textit{D. Jerison} [Geom. Funct. Anal. 13, No. 5, 1029--1081 (2003; Zbl 1055.35098)]; \textit{D. Jerison} and \textit{C. E. Kenig} [Ann. Math. (2) 121, 463--494 (1985; Zbl 0593.35119)]; \textit{C. E. Kenig} and \textit{N. Nadirashvili} [Math. Res. Lett. 7, No. 5--6, 625--630 (2000; Zbl 0973.35064)]; \textit{H. Koch} and \textit{D. Tataru} [J. Reine Angew. Math. 542, 133--146 (2002; Zbl 1222.35050)]; \textit{I. Rodnianski} and \textit{W. Schlag} [Invent. Math. 155, No. 3, 451--513 (2004; Zbl 1063.35035)]). Behavior of eigenvalues of certain Schrödinger operators in the rational Dunkl setting https://zbmath.org/1472.35259 2021-11-25T18:46:10.358925Z "Hejna, Agnieszka" https://zbmath.org/authors/?q=ai:hejna.agnieszka Summary: For a normalized root system $$R$$ in $$\mathbb{R}^N$$ and a multiplicity function $$k\ge 0$$ let $$\mathbf{N}=N+\sum_{\alpha \in R} k(\alpha)$$. We denote by $$dw(\mathbf{x})=\prod_{\alpha \in R}|\langle\mathbf{x},\alpha \rangle |^{k(\alpha)}d\mathbf{x}$$ the associated measure in $$\mathbb{R}^N$$. Let $$L=-\varDelta +V, V\ge 0$$, be the Dunkl-Schrödinger operator on $$\mathbb{R}^N$$. Assume that there exists $$q >\max (1,\frac{\mathbf{N}}{2})$$ such that $$V$$ belongs to the reverse Hölder class $$\mathrm{RH}^q(dw)$$. For $$\lambda>0$$ we provide upper and lower estimates for the number of eigenvalues of $$L$$ which are less or equal to $$\lambda$$. Our main tool in the Fefferman-Phong type inequality in the rational Dunkl setting. Decay estimates for three-dimensional Navier-Stokes equations with damping https://zbmath.org/1472.35274 2021-11-25T18:46:10.358925Z "Zhao, Xiaopeng" https://zbmath.org/authors/?q=ai:zhao.xiaopeng The Navier-Stokes system with the damping term $$|u|^{\beta-1}u$$ is studied in the whole space $${\mathbb R}^3$$. Decay rates for solutions of the Cauchy problem are derived, together with some auxiliary estimates of negative order Sobolev norms. Remarks on blow-up criteria for the derivative nonlinear Schrödinger equation under the optimal threshold setting https://zbmath.org/1472.35364 2021-11-25T18:46:10.358925Z "Takaoka, Hideo" https://zbmath.org/authors/?q=ai:takaoka.hideo Summary: We study the Cauchy problem of the mass critical nonlinear Schrödinger equation with derivative with the $$4 \pi$$ mass. One has the global well-posedness in $$H^1$$ whenever the mass is strictly less than $$4 \pi$$'' or whenever the mass is equal to $$4 \pi$$ and the momentum is strictly less than zero''. In this paper, by the concentration compact principle as originally done by \textit{C. E. Kenig} and \textit{F. Merle} [Invent. Math. 166, No. 3, 645--675 (2006; Zbl 1115.35125)], we obtain the limiting profile of blow up solutions with the critical $$4 \pi$$ mass. Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities https://zbmath.org/1472.37062 2021-11-25T18:46:10.358925Z "Zhu, Mengkun" https://zbmath.org/authors/?q=ai:zhu.mengkun "Wang, Dan" https://zbmath.org/authors/?q=ai:wang.dan "Chen, Yang" https://zbmath.org/authors/?q=ai:chen.yang.1|chen.yang.2 Summary: In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles $w(z,t)=A\theta (z-t)e^{-z^2+tz},$ here $$\theta(x)$$ is the Heaviside function, where $$A> 0$$, $$t>0$$, and $$z\in [0,\infty)$$. We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities $$R_n(t)$$ and $$r_n(t)$$ satisfy the coupled Riccati equations, from which we also prove that $$R_n(t)$$ satisfies a particular Painlevé IV equation. Even more, $$\sigma_n(t)$$, allied to $$R_n(t)$$, satisfies both the discrete and continuous Jimbo-Miwa-Okamoto $$\sigma$$-form of the Painlevé IV equation. Finally, we consider the situation when $$n$$ gets large, the second order linear differential equation satisfied by the polynomials $$P_n(x)$$ orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation. Discrete harmonic analysis associated with ultraspherical expansions https://zbmath.org/1472.39008 2021-11-25T18:46:10.358925Z "Betancor, Jorge J." https://zbmath.org/authors/?q=ai:betancor.jorge-j "Castro, Alejandro J." https://zbmath.org/authors/?q=ai:castro.alejandro-j "Fariña, Juan C." https://zbmath.org/authors/?q=ai:farina.juan-carlos "Rodríguez-Mesa, L." https://zbmath.org/authors/?q=ai:rodriguez-mesa.lourdes Authors' abstract: In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted $$\ell^p$$-boundedness properties of maximal operators and Littlewood-Paley $$g$$-functions defined by Poisson and heat semigroups generated by the difference operator $\Delta_{\lambda} f(n):=a_n^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \,\lambda >0,$ where $$a_n^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}$$, $$n\in \mathbb{N}$$, and $$a_{-1}^{\lambda }:=0$$. We also prove weighted $$\ell^p$$-boundedness properties of transplantation operators associated with the system $$\{\varphi_n^{\lambda } \}_{n\in \mathbb{N}}$$ of ultraspherical functions, a family of eigenfunctions of $$\Delta_\lambda$$. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting. Hypercontractivity of the semigroup of the fractional Laplacian on the $$n$$-sphere https://zbmath.org/1472.39048 2021-11-25T18:46:10.358925Z "Frank, Rupert L." https://zbmath.org/authors/?q=ai:frank.rupert-l "Ivanisvili, Paata" https://zbmath.org/authors/?q=ai:ivanisvili.paata The paper presents a further contribution to a problem concerning the hypercontractivity of the Poisson semigroup of $$e^{-t\sqrt{-\Delta}}$$ from $$L^p(\mathbb{S}^n)$$ to $$L^q(\mathbb{S}^n)$$ for $$t>0$$ on the sphere $$\mathbb{S}^n$$ of dimension $$n$$. This question was posed by \textit{C. E. Mueller} and \textit{F. B. Weissler} [J. Funct. Anal. 48, 252--283 (1982; Zbl 0506.46022)]. The main result of the paper states that for $$1<p\leq q$$ the condition $$e^{-t\sqrt{n}}\leq \sqrt{\frac{p-1}{q-1}}$$, carrying the smallest nonzero eigenvalue $$\sqrt{n}$$ of $$\sqrt{-\Delta}$$, is necessary and sufficient for dimension $$n\leq 3$$. Noteworthy, in case of $$q>\max\{2,p\}$$ the aforementioned condition is not sufficient for dimension $$n\geq 4$$. The reason of the exceptionality for $$n=1,2,3$$ is explained in detail in Subsection 2.2. In the remaining part of the paper it is proved, by contradiction, why the sufficient condition does not hold in general. Summing up, the question of finding an hypercontractivity equivalence for the Poisson semigroup on $$t>0$$ for $$n\geq 4$$ remains open. On the explicit representation of polyorthogonal polynomials https://zbmath.org/1472.41008 2021-11-25T18:46:10.358925Z "Starovoitov, A. P." https://zbmath.org/authors/?q=ai:starovoitov.aleksandr-pavlovich In the classical theory of orthogonal polynomials, the representation of the orthogonal polynomial $$Q_n$$ in determinant form is well known: $Q_n(z)=\left| \begin{array}{cccc}s_0 & s_1 & \cdots & s_n \\ s_1 & s_2 & \cdots & s_{n+1} \\ \cdots & \cdots & \cdots & \cdots \\ s_{n-1} & s_n & \cdots & s_{2n-1}\\ 1 & z & \cdots & z^n\end{array}\right|,$ where $s_i=\int_{\triangle}x^id\mu (x)\ (i=0, 1, \ldots )$ is a sequence of power moments of the measure $$\mu$$, whose carrier coincides with the orthogonality segment $$\triangle$$. In this paper, the author proves that a similar representation is also valid for a polyorthogonal polynomial if the power moments of the measures $$\mu _j$$ are known. In this respect, notice that polyorthogonal polynomials (named also as \lq multiple orthogonal polynomial\rq) satisfy non-standard orthogonality relations and naturally arise as the common denominator of the Padé approximations for the system of Markov functions. Here, the orthogonality relations are distributed among several measures $$\mu _1,...,\mu _k$$. Moreover, the author considers the concept of orthogonality in a broader sense, when the orthogonal polynomial is defined for an arbitrary sequence $$\{s_i\}$$, which is not bound by any restrictions. In this general situation, necessary and sufficient conditions for the existence and uniqueness of a polyorthogonal polynomial are found, under which it can be represented in determinant form similar to one above. As a consequence, similar representations are obtained for the residual function, the numerator, and the denominator of the corresponding Padé approximations, complementing this way well-known results in the theory of polyorthogonal polynomials and Padé approximations. A fundamental role is played by new concepts which are introduced in the paper, namely, an admissible index and an almost perfect system of functions. Correction to: Some trigonometric polynomials with extremely small uniform norm and their applications'' https://zbmath.org/1472.42001 2021-11-25T18:46:10.358925Z Correction to the article [\textit{A. O. Radomskii}, Izv. Math. 84, No. 2, 361--391 (2020; Zbl 1440.42002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 84, No. 2, 166--196 (2020)]. Hilbert transforms and the equidistribution of zeros of polynomials https://zbmath.org/1472.42002 2021-11-25T18:46:10.358925Z "Carneiro, Emanuel" https://zbmath.org/authors/?q=ai:carneiro.emanuel "Das, Mithun Kumar" https://zbmath.org/authors/?q=ai:das.mithun-kumar "Florea, Alexandra" https://zbmath.org/authors/?q=ai:florea.alexandra-m "Kumchev, Angel V." https://zbmath.org/authors/?q=ai:kumchev.angel-v "Malik, Amita" https://zbmath.org/authors/?q=ai:malik.amita "Milinovich, Micah B." https://zbmath.org/authors/?q=ai:milinovich.micah-b "Turnage-Butterbaugh, Caroline" https://zbmath.org/authors/?q=ai:turnage-butterbaugh.caroline-l "Wang, Jiuya" https://zbmath.org/authors/?q=ai:wang.jiuya Summary: We improve the current bounds for an inequality of \textit{P. Erdős} and \textit{P. Turán} [Ann. Math. (2) 51, 105--119 (1950; Zbl 0036.01501)] related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of \textit{K. Soundararajan} [Am. Math. Mon. 126, No. 3, 226--236 (2019; Zbl 1409.42003)], we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan [loc. cit.], refinements of the discrepancy inequality of Erdős and Turán had been obtained by \textit{T. Ganelius} [Ark. Mat. 3, 1--50 (1954; Zbl 0055.06905)] and \textit{M. Mignotte} [C. R. Acad. Sci., Paris, Sér. I 315, No. 8, 907--911 (1992; Zbl 0773.31002)]. On the value of the widths of some classes of functions from $$L_2$$ https://zbmath.org/1472.42003 2021-11-25T18:46:10.358925Z "Langarshoev, M. R." https://zbmath.org/authors/?q=ai:langarshoev.mukhtor-ramazonovich The author obtains sharp inequalities of Jackson-Stechkin type between the best approximations of periodic differentiable functions by trigonometric polynomials and generalized moduli of continuity of $$m$$-th order in the space $$L_2$$. He also writes down exact values for various $$n$$-widths of the classes of functions defined by the generalized moduli of continuity of $$r$$-th derivative of functions form $$L_2$$. A $$\mathcal{C}^1$$ composite spline Hermite interpolant on the sphere https://zbmath.org/1472.42004 2021-11-25T18:46:10.358925Z "Bouhiri, Salah" https://zbmath.org/authors/?q=ai:bouhiri.salah "Lamnii, Abdellah" https://zbmath.org/authors/?q=ai:lamnii.abdelleh "Lamnii, Mohamed" https://zbmath.org/authors/?q=ai:lamnii.mohamed "Zidna, Ahmed" https://zbmath.org/authors/?q=ai:zidna.ahmed Summary: In this paper, we identified the sphere-like surface by a rectangular domain. We constructed a new interpolant on the sphere using the tensor product of the quadratic composite-spline interpolant and the third-order trigonometric composite-spline interpolant. This construction on the rectangular domain is described in detail, with the study and the proof of the error bounds of each interpolant. Furthermore, we present numerical examples to show the efficiency of this method. Characterization of Fourier transform of $$H$$-valued functions on the real line https://zbmath.org/1472.42005 2021-11-25T18:46:10.358925Z "Biswas, Md Hasan Ali" https://zbmath.org/authors/?q=ai:biswas.md-hasan-ali "Radha, Ramakrishnan" https://zbmath.org/authors/?q=ai:radha.ramakrishnan Summary: A characterization is obtained for the Fourier transform of functions belonging to $$\mathcal{\mathcal{L}}^2 ( \mathbb{R} , H )$$, where $$H$$ denotes a Hilbert $$C^\ast$$-module. But in the case of functions belonging to $$L^1 ( \mathbb{R} , H )$$ a similar result is proved when $$H$$ is a separable Hilbert space. Evaluation formula and approximation for Wiener integrals via the Fourier-type functional https://zbmath.org/1472.42006 2021-11-25T18:46:10.358925Z "Chung, Hyun Soo" https://zbmath.org/authors/?q=ai:chung.hyun-soo "Lee, Un Gi" https://zbmath.org/authors/?q=ai:lee.un-gi Summary: In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations. Harmonic analysis associated to the canonical Fourier Bessel transform https://zbmath.org/1472.42007 2021-11-25T18:46:10.358925Z "Dhaouadi, Lazhar" https://zbmath.org/authors/?q=ai:dhaouadi.lazhar "Sahbani, Jihed" https://zbmath.org/authors/?q=ai:sahbani.jihed "Fitouhi, Ahmed" https://zbmath.org/authors/?q=ai:fitouhi.ahmed Summary: The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator $$\Delta^{\mathbf{m}}_{\nu}$$ on the real line: We define the canonical Fourier Bessel transform $$\mathcal{F}^{\mathbf{m}}_{\nu}$$ and study some of its important properties. We prove a Riemann-Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for $$\mathcal{F}^{\mathbf{m}}_{\nu}$$. In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle. Sufficient conditions for the absolute convergence of Fourier integral in terms of weighted spaces https://zbmath.org/1472.42008 2021-11-25T18:46:10.358925Z "Kolomoitsev, Yu. S." https://zbmath.org/authors/?q=ai:kolomoitsev.yurii-s Summary: Sufficient conditions for the representation of functions as an absolutely convergent Fourier integral in terms of simultaneous behavior of norms'', of functions in weighted Besov spaces are obtained. Uncertainty principles for Wigner-Ville distribution associated with the linear canonical transforms https://zbmath.org/1472.42009 2021-11-25T18:46:10.358925Z "Li, Yong-Gang" https://zbmath.org/authors/?q=ai:li.yonggang "Li, Bing-Zhao" https://zbmath.org/authors/?q=ai:li.bingzhao "Sun, Hua-Fei" https://zbmath.org/authors/?q=ai:sun.huafei Summary: The Heisenberg uncertainty principle of harmonic analysis plays an important role in modern applied mathematical applications, signal processing and physics community. The generalizations and extensions of the classical uncertainty principle to the novel transforms are becoming one of the most hottest research topics recently. In this paper, we firstly obtain the uncertainty principle for Wigner-Ville distribution and ambiguity function associate with the linear canonical transform, and then the $$n$$-dimensional cases are investigated in detail based on the proposed Heisenberg uncertainty principle of the $$n$$-dimensional linear canonical transform. Uncertainty principles for the quadratic-phase Fourier transforms https://zbmath.org/1472.42010 2021-11-25T18:46:10.358925Z "Shah, Firdous A." https://zbmath.org/authors/?q=ai:shah.firdous-ahmad "Nisar, Kottakkaran S." https://zbmath.org/authors/?q=ai:nisar.kottakkaran-s "Lone, Waseem Z." https://zbmath.org/authors/?q=ai:lone.waseem-z "Tantary, Azhar Y." https://zbmath.org/authors/?q=ai:tantary.azhar-y Summary: The quadratic-phase Fourier transform (QPFT) is a recent addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this article, we formulate several classes of uncertainty principles for the QPFT. Firstly, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding QPFT. Secondly, we obtain some logarithmic and local uncertainty inequalities such as Beckner and Sobolev inequalities for the QPFT. Thirdly, we study several concentration-based uncertainty principles, including Nazarov's, Amrein-Berthier-Benedicks's, and Donoho-Stark's uncertainty principles. Finally, we conclude the study with the formulation of Hardy's and Beurling's uncertainty principles for the QPFT. Sparse domination via the helicoidal method https://zbmath.org/1472.42011 2021-11-25T18:46:10.358925Z "Benea, Cristina" https://zbmath.org/authors/?q=ai:benea.cristina "Muscalu, Camil" https://zbmath.org/authors/?q=ai:muscalu.camil Summary: Using exclusively the localized estimates upon which the helicoidal method was built by the authors, we show how sparse estimates can also be obtained. This approach yields a sparse domination for scalar and multiple vector-valued extensions of operators alike. We illustrate these ideas for an $$n$$-linear Fourier multiplier whose symbol is singular along a $$k$$-dimensional subspace of $$\Gamma=\{\xi_1+\cdots+\xi_{n+1}=0\}$$, where $$k < (n+1)/{2}$$, and for the variational Carleson operator. A comparison principle for convolution measures with applications https://zbmath.org/1472.42012 2021-11-25T18:46:10.358925Z "Oliveira e Silva, Diogo" https://zbmath.org/authors/?q=ai:oliveira-e-silva.diogo "Quilodrán, René" https://zbmath.org/authors/?q=ai:quilodran.rene Summary: We establish the general form of a geometric comparison principle for $$n$$-fold convolutions of certain singular measures in $$\mathbb{R}^d$$ which holds for arbitrary $$n$$ and $$d$$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [\textit{G. Brocchi} et al., Anal. PDE 13, No. 2, 477--526 (2020; Zbl 1435.35015)]. Multiplier theorems via martingale transforms https://zbmath.org/1472.42013 2021-11-25T18:46:10.358925Z "Bañuelos, Rodrigo" https://zbmath.org/authors/?q=ai:banuelos.rodrigo "Baudoin, Fabrice" https://zbmath.org/authors/?q=ai:baudoin.fabrice "Chen, Li" https://zbmath.org/authors/?q=ai:chen.li.2|chen.li.4|chen.li.5|chen.li.6|chen.li.7|chen.li.1|chen.li.3 "Sire, Yannick" https://zbmath.org/authors/?q=ai:sire.yannick Summary: We develop a new and general approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by \textit{P. R. Stinga} and \textit{J. L. Torrea} [Commun. Partial Differ. Equations 35, No. 10--12, 2092--2122 (2010; Zbl 1209.26013)], which is of independent interest. Our methods here also recover the sharp $$L^p$$ bounds for second order Riesz transforms by a limiting argument. Some remarks on the Mikhlin-Hörmander and Marcinkiewicz multiplier theorems: a short historical account and a recent improvement https://zbmath.org/1472.42014 2021-11-25T18:46:10.358925Z "Grafakos, Loukas" https://zbmath.org/authors/?q=ai:grafakos.loukas In this paper under review, the author first presents a short historical overview of the Mikhlin-Hörmander and Marcinkiewicz multiplier theorems. Then, he considers different versions of them and provide comparisons. He also presents a recent improvement of the Marcinkiewicz multiplier theorem in the two-dimensional case. The boundedness of commutators of generalized fractional integral operators on specific generalized Morrey spaces https://zbmath.org/1472.42015 2021-11-25T18:46:10.358925Z "Budhi, Wono Setya" https://zbmath.org/authors/?q=ai:setya-budhi.wono|budhi.wono-setya "Lindiarni, Janny" https://zbmath.org/authors/?q=ai:lindiarni.janny Summary: In this note, we prove the boundedness of commutators of generalized fractional integral operators on the specific generalized Morrey spaces with different growth of functions. We call it specific Morrey spaces because the growth function relating with the kernel of integral operators. $$L^p$$ bounds for the commutators of oscillatory singular integrals with rough kernels https://zbmath.org/1472.42016 2021-11-25T18:46:10.358925Z "Chen, Yanping" https://zbmath.org/authors/?q=ai:chen.yanping.1 "Zhu, Kai" https://zbmath.org/authors/?q=ai:zhu.kai Summary: We establish the $$L^p$$ boundedness for some commutators of oscillatory singular integrals with the kernel condition which was introduced by \textit{L. Grafakos} and \textit{A. Stefanov} [Indiana Univ. Math. J. 47, No. 2, 455--469 (1998; Zbl 0913.42014)]. Our theorems contain various conditions on the phase function. Boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces https://zbmath.org/1472.42017 2021-11-25T18:46:10.358925Z "Fu, Zunwei" https://zbmath.org/authors/?q=ai:fu.zunwei "Lu, Shanzhen" https://zbmath.org/authors/?q=ai:lu.shanzhen "Pan, Yibiao" https://zbmath.org/authors/?q=ai:pan.yibiao "Shi, Shaoguang" https://zbmath.org/authors/?q=ai:shi.shaoguang Summary: We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures. Multilinear commutators of Calderón-Zygmund operator on generalized weighted Morrey spaces https://zbmath.org/1472.42018 2021-11-25T18:46:10.358925Z "Guliyev, Vagif S." https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir "Alizadeh, Farida Ch." https://zbmath.org/authors/?q=ai:alizadeh.farida-ch Summary: The boundedness of multilinear commutators of Calderón-Zygmund operator $$T_{\vec{b}}$$ on generalized weighted Morrey spaces $$M_{p,\varphi}(w)$$ with the weight function $$w$$ belonging to Muckenhoupt's class $$A_p$$ is studied. When $$1<p<\infty$$ and $$\vec{b}=(b_1, \dots, b_m)$$, $$b_i \in \mathrm{BMO}$$, $$i=1,\dots, m$$, the sufficient conditions on the pair $$(\varphi_1,\varphi_2)$$ which ensure the boundedness of the operator $$T_{\vec{b}}$$ from $$M_{p,\varphi_1}(w)$$ to $$M_{p,\varphi_2}(w)$$ are found. In all cases the conditions for the boundedness of $$T_{\vec{b}}$$ are given in terms of Zygmund-type integral inequalities on $$(\varphi_1,\varphi_2)$$, which do not assume any assumption on monotonicity of $$\varphi_1(x,r)$$, $$\varphi_2(x,r)$$ in $$r$$. On the compactness of oscillation and variation of commutators https://zbmath.org/1472.42019 2021-11-25T18:46:10.358925Z "Guo, Weichao" https://zbmath.org/authors/?q=ai:guo.weichao "Wen, Yongming" https://zbmath.org/authors/?q=ai:wen.yongming "Wu, Huoxiong" https://zbmath.org/authors/?q=ai:wu.huoxiong "Yang, Dongyong" https://zbmath.org/authors/?q=ai:yang.dongyong The singular integral operator with homogeneous kernel is defined by $T_{\Omega}f(x):=p.v.\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}f(y)dy,$ where $$\Omega$$ is a homogeneous function of degree zero and satisfies the following mean value zero property: $\int_{S^{n-1}}\Omega(x^\prime)d_{\sigma}(x^\prime)=0,$ where $$d_{\sigma}$$ is the spherical measure on the sphere $$S^{n-1}$$. Given a locally integrable function $$b$$ and a linear operator $$T$$, the commutator $$[b,T]$$ is defined by: $T^{b}(f)(x):=[b,T]f(x):=b(x)T(f)(x)-T(bf)(x)$ for suitable functions $$f$$. The paper is devoted to the weighted $$L_{p}$$-compactness of the oscillation and variation of the commutator of singular integral operator. The authors first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, they establish a new $$CMO(\mathbb{R}^{n})$$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces. The commutators of fractional integrals on generalized Herz spaces https://zbmath.org/1472.42020 2021-11-25T18:46:10.358925Z "Hu, Yue" https://zbmath.org/authors/?q=ai:hu.yue "He, Yuexiang" https://zbmath.org/authors/?q=ai:he.yuexiang "Wang, Yueshan" https://zbmath.org/authors/?q=ai:wang.yueshan Summary: Let $$I_l$$ be the fractional integral, $$0 < l < n$$ and let $$b \in \mathrm{BMO}$$. We will obtain the weighted estimates for the commutator $$[n, I_l]$$ on the generalized Herz spaces. On integral operators in weighted grand Lebesgue spaces of Banach-valued functions https://zbmath.org/1472.42021 2021-11-25T18:46:10.358925Z "Kokilashvili, Vakhtang" https://zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m "Meskhi, Alexander" https://zbmath.org/authors/?q=ai:meskhi.alexander Summary: The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case, we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator, we rely on the reasonable bound of the sharp constant in the Buckley-type theorem, which is also derived in the paper. $$L^2$$-bounded singular integrals on a purely unrectifiable set in $$R^d$$ https://zbmath.org/1472.42022 2021-11-25T18:46:10.358925Z "Mateu, Joan" https://zbmath.org/authors/?q=ai:mateu.joan "Prat, Laura" https://zbmath.org/authors/?q=ai:prat.laura Summary: We construct an example of a purely unrectifiable measure $$\mu$$ in $$\mathbb{R}^d$$ for which the singular integrals associated to the kernels $$K(x)=P_{2k+1}(x)/|x|^{2k+d}$$, with $$k\geq 1$$ and $$P_{2k+1}$$ a homogeneous harmonic polynomial of degree $$2k+1$$, are bounded in $$L^2(\mu)$$. This contrasts starkly with the results concerning the Riesz kernel $$x/|x|^d$$ in $$\mathbb{R}^d$$. Control of the bilinear indicator cube testing property https://zbmath.org/1472.42023 2021-11-25T18:46:10.358925Z "Sawyer, Eric T." https://zbmath.org/authors/?q=ai:sawyer.eric-t "Uriarte-Tuero, Ignacio" https://zbmath.org/authors/?q=ai:uriarte-tuero.ignacio Summary: We show that the $$\alpha$$-fractional bilinear indicator/cube testing constant $\mathcal{BICT}_{\text{T}^{\alpha}}(\sigma,\omega)\equiv\sup\limits_{Q\in\mathcal{P}^n}\sup\limits_{E,F\subset Q}\frac{1}{\sqrt{\left\vert Q\right\vert_{\sigma}\left\vert Q\right\vert_{\omega}}}\left\vert \int_FT_{\sigma}^{\alpha}\left(1_E\right)\omega \right\vert,$ defined for any $$\alpha$$-fractional singular integral $$\text{T}^{\alpha}$$ on $$\mathbb{R}^n$$ with $$0<\alpha< n$$, is controlled by the classical $$\alpha$$-fractional Muckenhoupt constant $$A_2^{\alpha}\left(\sigma,\omega\right)$$, provided the product measure $$\sigma\times\omega$$ is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding $$2\left(n-\alpha \right)$$. Moreover, this control is sharp within the class of diagonally reverse doubling product measures. In fact, every product measure $$\mu\times\mu$$, where $$\mu$$ is an Ahlfors-David regular measure $$\mu$$ with exponent $$n-\alpha$$, has diagonal exponent $$2\left( n-\alpha\right)$$ and satisfies $$A_2^{\alpha}\left( \mu ,\mu \right)<\infty$$ and $$\mathcal{BICT}_{I^{\alpha}}\left(\mu,\mu\right)=\infty$$, which has implications for the $$L^2$$ trace inequality of the fractional integral $$I^{\alpha}$$ on domains with fractional boundary. When combined with the main results in [\textit{E. T. Sawyer}, A T1 theorem for general Calderón-Zygmund operators with comparable doubling weights, and optimal cancellation conditions'', Preprint, \url{arXiv:1906.05602}, A restricted weak type inequality with application to a $$\text{T}_p$$ theorem and optimal cancellation conditions for CZO's'', Preprint, \url{arXiv:1907.07571}, T1 testing implies $$\text{T}_p$$ polynomial testing: optimal cancellation conditions for CZO's'', Preprint, \url{arXiv:1907.10734}] the above control of $$\mathcal{BICT}_{T^{\alpha}}$$ for $$\alpha> 0$$ yields a $$T1$$ theorem for doubling weights with appropriate diagonal reverse doubling, i.e. the norm inequality for $$\text{T}^{\alpha}$$ is controlled by cube testing constants and the $$\alpha$$-fractional one-tailed Muckenhoupt constants $$\mathcal{A}_2^{\alpha}$$ (without any energy assumptions), and also yields a corresponding cancellation condition theorem for the kernel of $$\text{T}^{\alpha}$$, both of which hold for arbitrary $$\alpha$$-fractional Calderón-Zygmund operators $$\text{T}^{\alpha}$$. We do not know if the analogous result for $$\mathcal{BICT}_H\left(\sigma,\omega\right)$$ holds for the Hilbert transform $$H$$ in case $$\alpha=0$$, but we show that $$\mathcal{BICT}_{H^{\mathrm{dy}}}\left(\sigma,\omega\right)$$ is not controlled by the Muckenhoupt condition $$\mathcal{A}_2^{\alpha}\left(\omega,\sigma\right)$$ for the dyadic Hilbert transform $$H^{\mathrm{dy}}$$ and doubling weights $$\sigma ,\omega$$. Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents https://zbmath.org/1472.42024 2021-11-25T18:46:10.358925Z "Wang, Hongbin" https://zbmath.org/authors/?q=ai:wang.hongbin.1|wang.hongbin "Xu, Jingshi" https://zbmath.org/authors/?q=ai:xu.jingshi "Tan, Jian" https://zbmath.org/authors/?q=ai:tan.jian.2|tan.jian|tan.jian.1 The authors prove boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents (see Theorem 2.1). As applications, for $$b=(b_1,\dots, b_m)$$ such that every $$b_i$$ is in the centeral BMO space with variable exponents, the authors further obtain boundedness of the multilinear Calderón-Zygmund commutator $$[\mathfrak{b}, T]$$ and $$T_{\mathfrak{b}}$$ on the product of central Morrey spaces with variable exponents (see Theorems 3.1 and 3.2), where $[\mathfrak{b}, T]f(x)= \int_{(\mathbb R^n)^m} K(x, y_1, \dots, y_m)\prod_{i=1}^m (b_i(x)-b_i(y_i)) f_i(y_i)\, dy_1\cdots\, dy_m$ and $T_{\mathfrak{b}}f(x)=\int_{\mathbb R^n} \prod_{i=1}^m (b_i(x)-b_i(y))K(x,y)f(y)\, dy.$ Commutator theorems for fractional integral operators on weighted Morrey spaces https://zbmath.org/1472.42025 2021-11-25T18:46:10.358925Z "Wang, Zhiheng" https://zbmath.org/authors/?q=ai:wang.zhiheng "Si, Zengyan" https://zbmath.org/authors/?q=ai:si.zengyan Summary: Let $$L$$ be the infinitesimal generator of an analytic semigroup on $$L^2(\mathbb{R}^n)$$ with Gaussian kernel bounds, and let $$L^{- \alpha / 2}$$ be the fractional integrals of $$L$$ for $$0 < \alpha < n$$. For any locally integrable function $$b$$, the commutators associated with $$L^{- \alpha / 2}$$ are defined by $$[b, L^{- \alpha / 2}](f)(x) = b(x) L^{- \alpha / 2}(f)(x) - L^{- \alpha / 2}(b f)(x)$$. When $$b \in \text{B} \text{M} \text{O}(\omega)$$ (weighted $$\text{B} \text{M} \text{O}$$ space) or $$b \in \text{B} \text{M} \text{O}$$, the authors obtain the necessary and sufficient conditions for the boundedness of $$[b, L^{- \alpha / 2}]$$ on weighted Morrey spaces, respectively. Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces https://zbmath.org/1472.42026 2021-11-25T18:46:10.358925Z "Zhang, Xiao" https://zbmath.org/authors/?q=ai:zhang.xiao "Liu, Feng" https://zbmath.org/authors/?q=ai:liu.feng.4|liu.feng|liu.feng.3|liu.feng.2|liu.feng.5|liu.feng.1 "Zhang, Huiyun" https://zbmath.org/authors/?q=ai:zhang.huiyun Summary: This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels $$\varOmega \in L^q (S^{n-1})$$ $$(q > 1)$$ and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces. Lipschitz spaces and fractional integral operators associated with nonhomogeneous metric measure spaces https://zbmath.org/1472.42027 2021-11-25T18:46:10.358925Z "Zhou, Jiang" https://zbmath.org/authors/?q=ai:zhou.jiang "Wang, Dinghuai" https://zbmath.org/authors/?q=ai:wang.dinghuai Summary: The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator from $$L^p \left(\mu\right)$$ into the space $$L^{q, \infty} \left(\mu\right)$$. Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented. Partitions of flat one-variate functions and a Fourier restriction theorem for related perturbations of the hyperbolic paraboloid https://zbmath.org/1472.42028 2021-11-25T18:46:10.358925Z "Buschenhenke, Stefan" https://zbmath.org/authors/?q=ai:buschenhenke.stefan "Müller, Detlef" https://zbmath.org/authors/?q=ai:muller.detlef "Vargas, Ana" https://zbmath.org/authors/?q=ai:vargas.ana-m|vargas.ana-lucia In this paper the authors consider problems associated to Fourier restriction to hyperbolic hypersurfaces, following upon their previous work found in [\textit{S. Buschenhenke} et al., On Fourier restriction for finite-type perturbations of the hyperbolic paraboloid'', Preprint, \url{arXiv:1902.05442}; Proc. Lond. Math. Soc. (3) 120, No. 1, 124--154 (2020; Zbl 1442.42044)]. Given a surface $$S$$, one may define the Fourier extension operator $$\mathscr{E}$$ by $\mathscr{E}f(\xi) = \widehat{f d\sigma}(\xi) = \int_S f(x) e^{-i\xi \cdot x} d\sigma(x)$ with $$f \in L^q(S, d\sigma)$$. Here $$\xi = (\xi_1, \xi_2, \xi_3) \in \mathbb{R}^3$$. The main result in this paper is the following: Assume that $$r > 10/3$$ and $$\frac{1}{q\prime} > \frac{2}{r}$$, and moreover suppose $$S$$ is the graph of the phase function $$\phi(x,y) = xy + h(y)$$, where the function $$h$$ is smooth and satisfies $h(0) = h^\prime(0) = h^{\prime\prime}(0) = 0\;.$ Assume further that either the function $$h$$ is of finite type at the origin, or flat and such that $$h^{\prime\prime\prime}(t)$$ is monotonic. Then if $$\Omega$$ is a sufficiently small neighborhood of the origin, $\left\|\mathscr{E}f\right\|_{L^r(\mathbb{R}^3)} \leq C_{r,q} \left\|f\right\|_{L^q(\Omega)}$ for all $$f \in L^q(\Omega)$$. Maximal operator on the space of continuous functions https://zbmath.org/1472.42029 2021-11-25T18:46:10.358925Z "Górka, Przemysław" https://zbmath.org/authors/?q=ai:gorka.przemyslaw Summary: We study the maximal operator on continuous functions in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Maximal operators and decoupling for $$\Lambda (p)$$ Cantor measures https://zbmath.org/1472.42030 2021-11-25T18:46:10.358925Z "Łaba, Isabella" https://zbmath.org/authors/?q=ai:laba.isabella Summary: For $$2\leq p<\infty$$, $$\alpha^\prime>2/p$$, and $$\delta>0$$, we construct Cantor-type measures on $$\mathbb{R}$$ supported on sets of Hausdorff dimension $$\alpha<\alpha'$$ for which the associated maximal operator is bounded from $$L^p_\delta (\mathbb{R})$$ to $$L^p(\mathbb{R})$$. Maximal theorems for fractal measures on the line were previously obtained by \textit{M. Pramanik} and \textit{I. Łaba} [Duke Math. J. 158, No. 3, 347--411 (2011; Zbl 1242.42011)]. The result here is weaker in that we are not able to obtain $$L^p$$ estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension $$\alpha>0$$, and have no Fourier decay. The proof is based on a decoupling inequality similar to that of \textit{I. Łaba} and \textit{H. Wang} [Int. Math. Res. Not. 2018, No. 9, 2944--2966 (2018; Zbl 1442.42031)]. On the regularity of the maximal function of a BV function https://zbmath.org/1472.42031 2021-11-25T18:46:10.358925Z "Lahti, Panu" https://zbmath.org/authors/?q=ai:lahti.panu Summary: We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that if the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of \textit{J. Weigt} [Variation of the uncentered maximal characteristic function'', Preprint, \url{arXiv:2004.10485}], we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function. Norm comparison estimates for the composite operator https://zbmath.org/1472.42032 2021-11-25T18:46:10.358925Z "Li, Xuexin" https://zbmath.org/authors/?q=ai:li.xuexin "Wang, Yong" https://zbmath.org/authors/?q=ai:wang.yong.10|wang.yong.9|wang.yong.8|wang.yong.7 "Xing, Yuming" https://zbmath.org/authors/?q=ai:xing.yuming Summary: This paper obtains the Lipschitz and BMO norm estimates for the composite operator $$\mathbb M_s \circ P$$ applied to differential forms. Here, $$\mathbb M_3$$ is the Hardy-Littlewood maximal operator, and $$P$$ is the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation. The dual conjecture of Muckenhoupt and Wheeden https://zbmath.org/1472.42033 2021-11-25T18:46:10.358925Z "Osękowski, Adam" https://zbmath.org/authors/?q=ai:osekowski.adam Summary: Let $$T$$ be a Calderón-Zygmund operator on $$\mathbb{R}^d$$. We prove the existence of a constant $$C_{T,d} < \infty$$ such that for any weight $$w$$ on $$\mathbb{R}^d$$ satisfying Muckenhoupt's condition $$A_1$$, we have $w\left(\{x\in \mathbb{R}^d:|Tf(x)| > w(x)\}\right) \leq C_{T,d}[w]_{A_1}\int_{\mathbb{R}^d}f \ \mathrm{d}x.$ The linear dependence on $$[w]_{A_1}$$, the $$A_1$$ characteristic of $$w$$, is optimal. The proof exploits the associated dimension-free inequalities for dyadic shifts. Another counterexample to Zygmund's conjecture https://zbmath.org/1472.42034 2021-11-25T18:46:10.358925Z "Rey, Guillermo" https://zbmath.org/authors/?q=ai:rey.guillermo It is well known that the Lebesgue differentiation theorem says that $\lim_{r\rightarrow0}\frac{1}{2r}\int_{x-r}^{x+r}f(y)dy=f(x)$ almost everywhere for all $$f\in L^{1}(\mathbb{R})$$. It is a natural question to study whether the basis of intervals can be replace for any other basis. We say that a basis $$\mathcal{B}$$ differentiates a function space $$X$$ if for all $$f\in X$$ $\lim_{k\rightarrow\infty}\frac{1}{|R_{k}|}\int_{R_{k}}f(y)dy=f(x)\qquad\text{a.e.}$ where $$\{R_{k}\}$$ is any sequence of elements in $$\mathcal{B}$$ containing $$x$$ and with diameters converging to $$0$$. The standard approach of this question reduces it to settling a weak-type estimate. In [Colloq. Math. 16, dedie a Franciszek Leja, 199--204 (1967; Zbl 0156.06301)] \textit{A. Zygmund} showed that the basis consisiting of $$d$$-dimensional intervals whose sides have no more than $$k$$ different sizes differentiates $$L(\log^{+}L)^{k-1}$$, namely that it satisfies a weak-type $$L(\log^{+}L)^{k-1}$$ $|\{x\in\mathbb{R}^{d}\,:\,\mathcal{\tilde{M}}f(x)>\lambda\}|\lesssim\int_{\mathbb{R}^{d}}\frac{|f(x)|}{\lambda}\log^{+}\left(\frac{|f(x)|}{\lambda}\right)^{k-1}dx\qquad\tag{*}$ where the maximal $$\tilde{\mathcal{M}}$$ function is defined taking supremum on a basis of rectangles of $$\mathbb{R}^{d}$$ in which rectangles have no more than $$k$$ different sideleghts. Zygmund's result leads to wonder whether if given $$d$$ $$\Phi_{i}:\mathbb{R}_{+}^{k}\rightarrow\mathbb{R}_{+}$$ non-decreasing functions in each variable, the basis consisting of all $$d$$-dimensional intervals $\Phi_{1}(t_{1},\dots,t_{k})\times\dots\times\Phi_{d}(t_{1},\dots,t_{k})$ differentiates $$L(\log^{+}L)^{k-1}.$$ Note that Zygmund's result settles a particular case of this conjecture. In [Bull. Am. Math. Soc., New Ser. 1, 255--257 (1979; Zbl 0431.42011)] \textit{A. Córdoba} settled the conjecture for the special case of intervals in $$\mathbb{R}^{3}$$ of dimensions $s\times t\times\Phi(s,t)$ for any function $$\Phi:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}_{+}$$ non-decreasing in each variable. \textit{F. Soria} [Ann. Math. (2) 123, 1--9 (1986; Zbl 0593.42007)] provided a counterexample in dimension three. He provides a basis of intervals of dimensions $s\times t\varphi(s)\times t\psi(s)$ with certain increasing functions $$\varphi$$ and $$\psi$$ which cannot differentiate $$L\log^{+}L$$ and also a basis of intervals of the form $s\times t\times\Phi_{1}(s,t)\times\Phi_{2}(s,t)$ that cannot either. In the paper under review counterexamples are provided for higher dimensions. It is shown that biparametric basis of intervals of the form $\Phi_{1}(s,t)\times\dots\times\Phi_{d}(s,t)$ for every $$d$$ for which $$(*)$$ only holds with $$k=d-1$$. Furthermore it is shown that in dimensions four and higher a basis with $$\Phi_{1}(s,t)=s$$ and $$\Phi_{2}(s,t)=t$$ such that $$(*)$$ only holds with $$k=d-1$$ can be provided as well. The approach used relies upon reducing the question to the dyadic setting via shifted dyadic rectangles. Endpoint regularity of the discrete multisublinear fractional maximal operators https://zbmath.org/1472.42035 2021-11-25T18:46:10.358925Z "Zhang, Xiao" https://zbmath.org/authors/?q=ai:zhang.xiao The main results of this article are about the discrete centered and uncentered $$m$$-sublinear fractional maximal operators, $$\mathfrak{M}_\alpha$$ and $$\widetilde{ \mathfrak{M} }_\alpha$$ (respectively). First, a theorem on endpoint regularity is obtained for $$\widetilde{ \mathfrak{M} }_\alpha$$ from the $$m$$-fold Cartesian product of $$\text{BV}(\mathbb{Z})$$ ($$0 \leq \alpha < 1$$) or $$\ell^1 (\mathbb{Z})$$ ($$m-1 \leq \alpha < m$$) into the space of functions with bounded $$q$$-variation, where $$q$$ depends on the subscript $$\alpha$$. Here $$\text{BV}(\mathbb{Z})$$ denotes the space of functions of bounded variation defined on $$\mathbb{Z}$$. After that, a theorem showing the boundedness of $$\mathfrak{M}_\alpha$$ and $$\widetilde{ \mathfrak{M} }_\alpha$$, $$0 \leq \alpha < m$$, from the $$m$$-fold Cartesian product of $$\ell^1 (\mathbb{Z})$$ into $$\text{BV}(\mathbb{Z})$$ is obtained. Both theorems can be considered as the discrete version of the corresponding results contained in [\textit{F. Liu} and \textit{H. Wu}, Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and also extend the corresponding already known results of \textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] and \textit{F. Liu} [Bull. Aust. Math. Soc. 95, No. 1, 108--120 (2017; Zbl 1364.42020)]. On the Helmholtz decompositions of vector fields of bounded mean oscillation and in real Hardy spaces over the half space https://zbmath.org/1472.42036 2021-11-25T18:46:10.358925Z "Giga, Yoshikazu" https://zbmath.org/authors/?q=ai:giga.yoshikazu "Gu, Zhongyang" https://zbmath.org/authors/?q=ai:gu.zhongyang Summary: This paper is concerned with the Helmholtz decompositions of vector fields of bounded mean oscillation over the half space and vector fields in real Hardy spaces over the half space. It proves the Helmholtz decomposition for vector fields of bounded mean oscillation over the half space whereas a partial Helmholtz decomposition for vector fields in real Hardy spaces over the half space. Meanwhile, it also establishes two sets of theories of real Hardy spaces over the half space which are compatible with the theory of \textit{A. Miyachi} [Stud. Math. 96, No. 3, 205--228 (1990; Zbl 0716.42017)]. Duality for outer $$L^p_\mu (\ell^r)$$ spaces and relation to tent spaces https://zbmath.org/1472.42037 2021-11-25T18:46:10.358925Z "Fraccaroli, Marco" https://zbmath.org/authors/?q=ai:fraccaroli.marco In this paper the author studies the outer $$L^p$$ spaces introduced by \textit{Y. Do} and \textit{C. Thiele} [Bull. Am. Math. Soc., New Ser. 52, No. 2, 249--296 (2015; Zbl 1318.42016)] on sets endowed with a measure and an outer measure. The author proves that in the case of finite sets, for $$1 < p \leq \infty$$, $$1 \leq r < \infty$$ or $$p=r \in \{ 1,\ \infty\}$$, the outer $$L_{\mu}^{p}(l^{r})$$ quasi-norm is equivalent to a norm, and $$L_{\mu}^{p}(l^r)$$ space is the Köthe dual space of $$L_{\mu}^{p^\prime}(l^{r^\prime})$$. The author also shows that in the upper half space setting the above properties hold true in the full range $$1 \leq p$$, $$r \leq \infty$$ and establishes the equivalence between the classical tent space $$T^p_r$$ and the outer $$L_{\mu}^{p}(l^r)$$ space in the upper half space. Furthermore, the author gives a full classification of weak and strong type estimates for a class of embedding maps from classical $$L^p$$ spaces in $$\mathbb{R}^n$$ to outer $$L^{p}(l^r)$$ spaces in the upper half space with a fractional scale factor. On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces https://zbmath.org/1472.42038 2021-11-25T18:46:10.358925Z "Guliyev, Vagif S." https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir "Deringoz, Fatih" https://zbmath.org/authors/?q=ai:deringoz.fatih Summary: We consider generalized Orlicz-Morrey spaces $$M_{\Phi,\varphi}(\mathbb R^n)$$ including their weak versions $$WM_{\Phi,\varphi}(\mathbb R^n)$$. In these spaces we prove the boundedness of the Riesz potential from $$M_{\Phi,\varphi_1}(\mathbb R^n)$$ to $$M_{\Psi,\varphi_2}(\mathbb R^n)$$ and from $$M_{\Phi,\varphi_1}(\mathbb R^n)$$ to $$WM_{\Psi,\varphi_2}(\mathbb R^n)$$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $$(\varphi_1, \varphi_2)$$, which do not assume any assumption on monotonicity of $$\varphi_1(x, r)$$, $$\varphi_2(x, r)$$ in $$r$$. Functions of nearly maximal Gowers-Host-Kra norms on Euclidean spaces https://zbmath.org/1472.42039 2021-11-25T18:46:10.358925Z "Neuman, A. Martina" https://zbmath.org/authors/?q=ai:neuman.a-martina Summary: Let $$k\ge 2$$, $$n\ge 1$$ be integers. Let $$f: \mathbb{R}^n \rightarrow\mathbb{C}$$. The $$k$$-th Gowers-Host-Kra norm of $$f$$ is defined recursively by $\Vert f\Vert_{U^k}^{2^k} =\int_{\mathbb{R}^n} \Vert T^hf \cdot{\bar{f}} \Vert_{U^{k-1}}^{2^{k-1}} \, \mathrm{d}h$ with $$T^hf(x) = f(x+h)$$ and $$\Vert f\Vert_{U^1} = | \int_{\mathbb{R}^n} f(x)\, \mathrm{d}x |$$. These norms were introduced by \textit{W. T. Gowers} [Geom. Funct. Anal. 11, No. 3, 465--588 (2001; Zbl 1028.11005)] in his work on Szemerédi's theorem, and by \textit{B. Host} and \textit{B. Kra} [Ann. Math. (2) 161, No. 1, 397--488 (2005; Zbl 1077.37002)] in ergodic setting. These norms are also discussed extensively in \textit{T. Tao} and \textit{V. H. Vu} [Additive combinatorics. Cambridge: Cambridge University Press (2006; Zbl 1127.11002)]. It is shown by \textit{T. Eisner} and \textit{T. Tao} [J. Anal. Math. 117, 133--186 (2012; Zbl 1305.11009)] that for every $$k\ge 2$$ there exist $$A(k,n)< \infty$$ and $$p_k = 2^k/(k+1)$$ such that $$\Vert f\Vert_{U^k} \le A(k,n)\Vert f\Vert_{p_k}$$, for all $$f \in L^{p_k}(\mathbb{R}^n)$$. The optimal constant $$A(k, n)$$ and the extremizers for this inequality are known [Eisner and Tao, loc. cit.]. In this dissertation, it is shown that if the ratio $$\Vert f \Vert_{U^k}/\Vert f\Vert_{p_k}$$ is nearly maximal, then $$f$$ is close in $$L^{p_k}$$ norm to an extremizer. On uniqueness sets for Walsh-Paley series https://zbmath.org/1472.42040 2021-11-25T18:46:10.358925Z "Kozlovskaya, T. D." https://zbmath.org/authors/?q=ai:kozlovskaya.t-d Summary: We classify $$U_p$$-sets for the Walsh system depending on the interval of all values of $$p$$ for which a given set is a $$U_p$$-set. On the Schur-Horn problem https://zbmath.org/1472.42041 2021-11-25T18:46:10.358925Z "Abtahi, Fatemeh" https://zbmath.org/authors/?q=ai:abtahi.fatemeh "Kamali, Zeinab" https://zbmath.org/authors/?q=ai:kamali.zeinab "Keyshams, Zahra" https://zbmath.org/authors/?q=ai:keyshams.zahra Summary: Let $$\mathcal{H}$$ be a separable Hilbert space. Recently, the concept of $$K$$-$$g$$-frame was introduced as a special generalization of $$g$$-Bessel sequences. In this paper, we point out some gaps in the proof of some existent results concerning $$K$$-$$g$$-frame. We present examples to indicate that these results are not necessarily valid. Then we remove the gaps and provide some desired conclusions. In this respect, we deal with Schur-Horn problem, which characterizes sequences $$\{\parallel f_n \parallel^2\}_{n = 1}^\infty$$, for all frames $$\{f_n\}_{n = 1}^\infty$$ with the same frame operator. We introduce the concept of synthesis related frames. Finally, as the main result, we investigate around Schur-Horn problem, for the case where $$\mathcal{H}$$ is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related. Inequalities for wavelet frames with composite dilations in $$L^2(\mathbb{R}^n)$$ https://zbmath.org/1472.42042 2021-11-25T18:46:10.358925Z "Ahmad, Owais" https://zbmath.org/authors/?q=ai:ahmad.owais "Sheikh, Neyaz A." https://zbmath.org/authors/?q=ai:sheikh.neyaz-ahmad Summary: We obtain generalized inequalities for wavelet frames with composite dilations by virtue of Fourier transform. Analysis of adaptive short-time Fourier transform-based synchrosqueezing transform https://zbmath.org/1472.42043 2021-11-25T18:46:10.358925Z "Cai, Haiyan" https://zbmath.org/authors/?q=ai:cai.haiyan "Jiang, Qingtang" https://zbmath.org/authors/?q=ai:jiang.qingtang "Li, Lin" https://zbmath.org/authors/?q=ai:li.lin.2|li.lin.1 "Suter, Bruce W." https://zbmath.org/authors/?q=ai:suter.bruce-w Continuous Schauder frames for Banach spaces https://zbmath.org/1472.42044 2021-11-25T18:46:10.358925Z "Eisner, Joseph" https://zbmath.org/authors/?q=ai:eisner.joseph "Freeman, Daniel" https://zbmath.org/authors/?q=ai:freeman.daniel-h-jun Continuous Schauder frames for a Banach space $$X$$ were introduced in this paper. This concept generalizes the concept of continuous frames for Hilbert spaces as well as that of unconditional Schauder frames for Banach spaces. It was proved that some basic properties remain to be true in this general setting. Several equivalent conditions were obtained for shrinking properties, and/or boundedly completeness of continuous Schauder frames. In particular, it was proved that the reflexivity of the Banach space $$X$$ is equivalent to the shrinking and boundedly completeness property of a continuous Schauder frame. Frame spectral pairs and exponential bases https://zbmath.org/1472.42045 2021-11-25T18:46:10.358925Z "Frederick, Christina" https://zbmath.org/authors/?q=ai:frederick.christina "Mayeli, Azita" https://zbmath.org/authors/?q=ai:mayeli.azita Summary: Given a domain $$\varOmega \subset\mathbb{R}^d$$ with positive and finite Lebesgue measure and a discrete set $$\varLambda \subset\mathbb{R}^d$$, we say that $$(\varOmega , \varLambda )$$ is a frame spectral pair if the set of exponential functions $$\mathcal{E}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}$$ is a frame for $$L^2(\varOmega )$$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $$\mathbb{Z}_N^d$$, $$d$$, $$N\ge 1$$, a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in $$\mathbb{R}^d$$ by adding'' a frame spectral pair in $$\mathbb{R}^d$$ to a frame spectral pair in $$\mathbb{Z}_N^d$$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory. A class of warped filter bank frames tailored to non-linear frequency scales https://zbmath.org/1472.42046 2021-11-25T18:46:10.358925Z "Holighaus, Nicki" https://zbmath.org/authors/?q=ai:holighaus.nicki "Wiesmeyr, Christoph" https://zbmath.org/authors/?q=ai:wiesmeyr.christoph "Průša, Zdeněk" https://zbmath.org/authors/?q=ai:prusa.zdenek The authors constructed a method for non-uniform filter banks starting from a uniform system of translates, generated by a prototype filter, a non-uniform covering of the frequency axis. The warping function is a $$C^1$$-diffeomorphism that determines the frequency progression and can be chosen freely. They combined with appropriately chosen decimation factors, a non-uniform analysis filter bank is obtained. They also constructed a filter bank adapted to a frequency scale derived from human auditory perception and families of filter banks that can be interpreted as an interpolation between linear (Gabor) and logarithmic (wavelet) frequency scales. A simple and constructive method for obtaining tight frames with bandlimited filters is derived by invoking previous results on generalized shift-invariant systems. Duality principles for $$F_a$$-frame theory in $$L^2 (\mathbb{R}_+ )$$ https://zbmath.org/1472.42047 2021-11-25T18:46:10.358925Z "Li, Yun-Zhang" https://zbmath.org/authors/?q=ai:li.yunzhang|li.yunzhang.1 "Hussain, Tufail" https://zbmath.org/authors/?q=ai:hussain.tufail Authors' abstract: The notion of R-dual in general Hilbert spaces was first introduced by \textit{P. G. Casazza} et al. [J. Fourier Anal. Appl. 10, No. 4, 383--408 (2004; Zbl 1058.42020)], with the motivation to obtain a general version of the duality principle in Gabor analysis. On the other hand, the space $$L^{2}(\mathbb{R}_{+})$$ of square integrable functions on the half real line $$\mathbb{R}_{+}$$ admits no traditional wavelet or Gabor frame due to $$\mathbb{R}_{+}$$ being not a group under addition. $$F_{a}$$-frame theory based on function-valued inner product'' is a new tool for analysis on $$L^{2}(\mathbb{R}_{+})$$. This paper addresses duality relations for $$F_{a}$$-frame theory in $$L^{2}(\mathbb{R}_{+})$$. We introduce the notion of $$F_{a}$$-R-dual of a given sequence in $$L^{2}(\mathbb{R}_{+})$$, and obtain some duality principles. Specifically, we prove that a sequence in $$L^{2}(\mathbb{R}_{+})$$ is an $$F_{a}$$-frame ($$F_{a}$$-Bessel sequence, $$F_{a}$$-Riesz basis, $$F_{a}$$-frame sequence) if and only if its $$F_{a}$$-R-dual is an $$F_{a}$$- Riesz sequence ($$F_{a}$$-Bessel sequence, $$F_{a}$$-Riesz basis, $$F_{a}$$-frame sequence), and that two sequences in $$L^{2}(\mathbb{R}_{+})$$ form a pair of $$F_{a}$$-dual frames if and only if their $$F_{a}$$-R-duals are $$F_{a}$$-biorthonormal. Full spark frames in the orbit of a representation https://zbmath.org/1472.42048 2021-11-25T18:46:10.358925Z "Malikiosis, Romanos Diogenes" https://zbmath.org/authors/?q=ai:malikiosis.romanos-diogenes "Oussa, Vignon" https://zbmath.org/authors/?q=ai:oussa.vignon-s|oussa.vignon The authors present a new infinite family of full spark equal norm tight frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. They illustrate the results by providing explicit constructions of full spark frames. Positive weight function and classification of $$g$$-frames https://zbmath.org/1472.42049 2021-11-25T18:46:10.358925Z "Poria, Anirudha" https://zbmath.org/authors/?q=ai:poria.anirudha The article under review characterized a class of $$g$$-frames, $$g$$-Riesz bases and $$g$$-orthonormal bases associated with a positive weight function and an isometry map on a Hilbert space. As application, the author applied this characterization to frames for shift-invariant subspaces on the Heisenberg group. Positive operator-valued measures and densely defined operator-valued frames https://zbmath.org/1472.42050 2021-11-25T18:46:10.358925Z "Robinson, Benjamin" https://zbmath.org/authors/?q=ai:robinson.benjamin-d "Moran, Bill" https://zbmath.org/authors/?q=ai:moran.william "Cochran, Doug" https://zbmath.org/authors/?q=ai:cochran.doug Summary: In the signal-processing literature, a frame is a mechanism for performing analysis and reconstruction in a Hilbert space. By contrast, in quantum theory, a positive operator-valued measure (POVM) decomposes a Hilbert-space vector for the purpose of computing measurement probabilities. Frames and their most common generalizations can be seen to give rise to POVMs, but does every reasonable POVM arise from a type of frame? We answer this question using a Radon-Nikodym-type result. Inequalities for nonuniform wavelet frames https://zbmath.org/1472.42051 2021-11-25T18:46:10.358925Z "Shah, Firdous A." https://zbmath.org/authors/?q=ai:shah.firdous-ahmad Summary: \textit{J.-P. Gabardo} and \textit{M. Z. Nashed} [J. Funct. Anal. 158, No. 1, 209--241 (1998; Zbl 0910.42018)] studied nonuniform wavelets by using the theory of spectral pairs for which the translation set $$\Lambda=\{0,r/N\}+2\mathbb{Z}$$ is no longer a discrete subgroup of $$\mathbb{R}$$ but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system $$\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^jx-\lambda),\,j\in\mathbb{Z},\, \lambda\in\Lambda\}$$ to be a frame for $$L^2(\mathbb{R})$$. The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system. Biorthogonal wavelets on the spectrum https://zbmath.org/1472.42052 2021-11-25T18:46:10.358925Z "Ahmad, Owais" https://zbmath.org/authors/?q=ai:ahmad.owais "Sheikh, Neyaz A." https://zbmath.org/authors/?q=ai:sheikh.neyaz-ahmad "Nisar, Kottakkaran Sooppy" https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran "Shah, Firdous A." https://zbmath.org/authors/?q=ai:shah.firdous-ahmad Given the set $$\Lambda=\{0,r/N\}+2\mathbb Z$$, where $$N\geq1$$ is an integer and $$r$$ is an odd integer such that $$1\leq r\leq2N-1$$, the authors establish necessary and sufficient condtions for the $$\Lambda$$-translates of a single function to form a Riesz basis for their closed linear span. They also introduce a notion of biorthogonal wavelts on $$\Lambda$$ and provide a complete characterization for the biorthogonality of translates of scaling functions of two NUMRAs (i.e., nonuniform mulitiresoltution analyses where the translation set acting on the scaling function associated with MRA is a union of $$\mathbb Z$$ and a translate of $$\mathbb Z$$) and the associated wavelet families. Additionally, they show that the nonuniform wavelets generate Riesz bases for $$L^2(\mathbb R)$$, provided that some mild assumptions on the scaling functions and the wavelets are satisfied. The paper is well organized, most of the necessary notions are introduced. The proofs are technical. Linguistic mistakes slightly disturb reading. On the unification of schemes and software for wavelets on the interval https://zbmath.org/1472.42053 2021-11-25T18:46:10.358925Z "Antun, Vegard" https://zbmath.org/authors/?q=ai:antun.vegard "Ryan, Øyvind" https://zbmath.org/authors/?q=ai:ryan.oyvind In this paper, the authors establish the fundaments for software supporting the recent constructions of wavelets on the interval of the form $$[0, M]$$, where $$M\in\mathbb{Z}$$. This includes extensions of some known cases of wavelets with various degrees of polynomial exactness. The paper is well-written and interesting which has undoubtedly reduced many tedious calculations involved in the software implementation accompanying the paper. A new fractional wavelet transform https://zbmath.org/1472.42054 2021-11-25T18:46:10.358925Z "Dai, Hongzhe" https://zbmath.org/authors/?q=ai:dai.hongzhe "Zheng, Zhibao" https://zbmath.org/authors/?q=ai:zheng.zhibao "Wang, Wei" https://zbmath.org/authors/?q=ai:wang.wei.30 The fractional Fourier transform and the continuous wavelet transform are useful in many applications. In this paper, using the fractional Fourier transform, the authors generalized the continuous wavelet transform. For a given signal $$x\in L^2(\mathbb{R})$$, its continuous wavelet transform $$\mbox{WT}_x$$ is defined to be $\mbox{WT}_x(a,b)=\langle x(t), \varphi_{a,b}(t)\rangle= \int_{-\infty}^\infty x(t) \overline{\varphi_{a,b}(t)}dt$ with $\varphi_{a,b}(t)=\frac{1}{\sqrt{a}} \varphi\left(\frac{t-b}{a}\right),\qquad t\in \mathbb{R}$ for $$a\in \mathbb{R}^+$$ and $$b\in \mathbb{R}$$, where $$\varphi$$ is an admissible mother wavelet satisfying $$\int_0^\infty |\omega|^{-1} |\Phi(\omega)|^2 d\omega<\infty$$ and $$\Phi$$ is the Fourier transform of $$\varphi$$. For $$\alpha\in \mathbb{R}$$, the proposed $$\alpha$$-order fractional wavelet transform of a given signal $$x\in L^2(\mathbb{R})$$ is defined to be $W_x^\alpha(a,b)=\langle x(t), \varphi_{\alpha,a,b}(t)\rangle= \int_{-\infty}^\infty x(y) \overline{\varphi_{\alpha,a,b}(t)}dt =\mbox{WT}_{\tilde{x}}(a,b)$ for $$a\in \mathbb{R}^+$$ and $$b\in \mathbb{R}$$, where $$\tilde{x}(t)= x(t)e^{\frac{j}{2}(t^2-b^2-(\frac{t-b}{a})^2)\cot \alpha}$$ and $\varphi_{\alpha,a,b}(t)=e^{-\frac{j}{2}(t^2-b^2-(\frac{t-b}{a})^2)\cot \alpha} \varphi_{a,b}(t)$ and here $$j$$ is the imaginary unit. Because $$\cot(\pi/2)=0$$, the $$\alpha$$-order fractional wavelet transform with $$\alpha=\pi/2$$ becomes the conventional continuous wavelet transform. Then the authors generalized many properties of the standard continuous wavelet transform to the $$\alpha$$-order fractional wavelet transform using the $$\alpha$$-order fractional Fourier transform. By discretizing the $$\alpha$$-order fractional wavelet transform, similar results for discrete wavelets such as the multiresolution analysis and orthogonal wavelets have been discussed for discretizing $$\alpha$$-order fractional wavelet transform. In particular, a Haar-type orthogonal fractional wavelet is constructed. A few examples are provided to illustrate the applications of the $$\alpha$$-order fractional wavelet transform to signal processing. Hexagonally symmetric orthogonal filters with $$\sqrt{3}$$ refinement https://zbmath.org/1472.42055 2021-11-25T18:46:10.358925Z "Krivoshein, Aleksandr" https://zbmath.org/authors/?q=ai:krivoshein.aleksandr-vladimirovich In wavelet theory, it is desired to have symmetry properties to work out well for applications. In the present paper, the author provides a way to parametrize orthogonal hexagonally symmetric low-pass filters with size up to 7-by-7 and also various symmetry centers. Then, he proposes a method to produce tight wavelet frames that inherit the symmetry from the initial low-pass filter. Finally, he suggests how to construct orthogonal hexagonally symmetric wavelets. High balanced biorthogonal multiwavelets with symmetry https://zbmath.org/1472.42056 2021-11-25T18:46:10.358925Z "Li, Youfa" https://zbmath.org/authors/?q=ai:li.youfa "Yang, Shouzhi" https://zbmath.org/authors/?q=ai:yang.shouzhi "Shen, Yanfeng" https://zbmath.org/authors/?q=ai:shen.yanfeng "Zhang, Gengrong" https://zbmath.org/authors/?q=ai:zhang.gengrong Summary: Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. In [Sci. China, Ser. F 49, No. 4, 504--515 (2006; Zbl 1129.42019); Sci. China, Ser. A 49, No. 1, 86--97 (2006; Zbl 1193.42100)], \textit{S. Yang} and \textit{L. Peng} constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results. Fast algorithms for function decomposition based on $$n$$-separate periodic wavelets https://zbmath.org/1472.42057 2021-11-25T18:46:10.358925Z "Pleshcheva, E. A." https://zbmath.org/authors/?q=ai:pleshcheva.e-a Summary: In this paper we give the definition and construction of a theory of periodic $$n$$-separate MRA and wavelets on the base of several scaling functions. We give effective numerical algorithms for decomposition of the function applying constructed periodic wavelets and scaling functions. For the entire collection see [Zbl 1467.34001]. Performance of reconstruction factors for a class of new complex continuous wavelets https://zbmath.org/1472.42058 2021-11-25T18:46:10.358925Z "Rayeezuddin, Mohammed" https://zbmath.org/authors/?q=ai:rayeezuddin.mohammed "Krishna Reddy, B." https://zbmath.org/authors/?q=ai:reddy.b-krishna "Sudheer Reddy, D." https://zbmath.org/authors/?q=ai:reddy.d-sudheer The continuous wavelet transform is an important tool in signal processing. For a given signal $$x\in L^2(\mathbb{R})$$, its continuous wavelet transform $$W^\psi_x$$ is defined to be $W^\psi_x(a,b)= \int_{-\infty}^\infty x(t) \frac{1}{\sqrt{a}} \overline{\psi\left(\frac{t-b}{a}\right)}dt, \qquad$ for $$a\in \mathbb{R}^+$$ and $$b\in \mathbb{R}$$, where $$\psi$$ is an admissible mother wavelet satisfying $$\int_{-\infty}^\infty \frac{|\hat{\psi}(\omega)|^2}{\omega} d\omega<\infty$$. The purpose of this paper is to use a family of new complex admissible wavelets $$\psi$$. More precisely, for any positive integer $$k$$, the authors considered the continuous wavelet transform by replacing the above wavelet $$\psi$$ with the complex-valued function $$\psi^k$$ given by $\psi^k(t)=(-1)^p \frac{1}{\sqrt{C_k}} \frac{d^k}{dt^k} (\theta(t)) \quad \mbox{with}\quad \theta(t)=\frac{e^{-it}}{1+t^2},$ where $$p=k/2$$ for even $$k$$ and $$p=(k+1)/2$$ for odd $$k$$, and $$C_k$$ is the normalization constant such that $$\|\psi^k\|=1$$. Examples and various quantities for such wavelets $$\psi^k$$ are provided in this paper. Then the authors discussed how to implement such complex continuous wavelet transform for discrete data on a bounded interval in Section 3 and how to reconstruct the given time series from its continuous wavelet transform in Section 5. As demonstrated through numerical examples in Section 5 for three examples of discrete signals, the reconstruction errors of the proposed new complex continuous wavelets are comparable with the Morlet, Paul, and DOG wavelets. In particular, the proposed complex wavelet $$\psi^2$$ (called crsw(2) in this paper) performs comparably with that of Morlet, Paul, and DOG wavelets. Generally, the reconstruction errors increase with respect to $$k$$ by using crsw(k) as $$k$$ increases. An improved uncertainty principle for functions with symmetry https://zbmath.org/1472.43004 2021-11-25T18:46:10.358925Z "Garcia, Stephan Ramon" https://zbmath.org/authors/?q=ai:garcia.stephan-ramon "Karaali, Gizem" https://zbmath.org/authors/?q=ai:karaali.gizem "Katz, Daniel J." https://zbmath.org/authors/?q=ai:katz.daniel-j The authors prove a generalization of a result by Chebotarëv which states that every minor of a discrete Fourier matrix of prime order is nonzero. A generalization of the Biró-Meshulam-Tao uncertainty principle [\textit{R.~Meshulam}, Eur. J. Comb. 27, No.~1, 63--67 (2006; Zbl 1145.43005); \textit{T.~Tao}, Math. Res. Lett. 12, No.~1, 121--127 (2005; Zbl 1080.42002)] to functions with symmetries that arise from certain group actions is used to establish the result. As special cases, the result includes analogues for discrete cosine and discrete sine matrices. The authors also show that their result is best possible and in some cases is stronger than that of Biró-Meshulam-Tao. Some of these results are shown to hold for non-prime fields under certain conditions. Equi-distributed property and spectral set conjecture on $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$ https://zbmath.org/1472.43007 2021-11-25T18:46:10.358925Z "Shi, Ruxi" https://zbmath.org/authors/?q=ai:shi.ruxi Summary: In this paper, we show an equi-distributed property in 2-dimensional finite abelian groups $$\mathbb{Z}_{p^n} \times \mathbb{Z}_{p^m}$$, where $$p$$ is a prime number. By using this equi-distributed property, we prove that Fuglede's spectral set conjecture holds on groups $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$, namely, a set in $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$ is a spectral set if and only if it is a translational tile. Integrability properties of integral transforms via Morrey spaces https://zbmath.org/1472.46031 2021-11-25T18:46:10.358925Z "Samko, Natasha" https://zbmath.org/authors/?q=ai:samko.natasha Summary: We show that integrability properties of integral transforms with kernel depending on the product of arguments (which include in particular, popular Laplace, Hankel, Mittag-Leffler transforms and various others) are better described in terms of Morrey spaces than in terms of Lebesgue spaces. Mapping properties of integral transforms of such a type in Lebesgue spaces, including weight setting, are known. We discover that local weighted Morrey and complementary Morrey spaces are very appropriate spaces for describing integrability properties of such transforms. More precisely, we show that under certain natural assumptions on the kernel, transforms under consideration act from local weighted Morrey space to a weighted complementary Morrey space and vice versa, where an interplay between behavior of functions and their transforms at the origin and infinity is transparent. In case of multidimensional integral transforms, for this goal we introduce and use anisotropic mixed norm Morrey and complementary Morrey spaces. Boundedness of localization operators on Lorentz mixed-normed modulation spaces https://zbmath.org/1472.46032 2021-11-25T18:46:10.358925Z "Sandıkçı, Ayşe" https://zbmath.org/authors/?q=ai:sandikci.ayse Summary: In this work we study certain boundedness properties for localization operators on Lorentz mixed-normed modulation spaces, when the operator symbols belong to appropriate modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms. A truncated real interpolation method and characterizations of screened Sobolev spaces https://zbmath.org/1472.46039 2021-11-25T18:46:10.358925Z "Stevenson, Noah" https://zbmath.org/authors/?q=ai:stevenson.noah "Tice, Ian" https://zbmath.org/authors/?q=ai:tice.ian Summary: In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization. Classic and exotic Besov spaces induced by good grids https://zbmath.org/1472.46041 2021-11-25T18:46:10.358925Z "Smania, Daniel" https://zbmath.org/authors/?q=ai:smania.daniel Summary: In a previous work we introduced Besov spaces $$\mathcal{B}^s_{p,q}$$ defined on a measure space with a good grid, with $$p\in[1,\infty)$$, $$q\in[1,\infty]$$ and $$0< s<1/p$$. Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces. Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis https://zbmath.org/1472.46044 2021-11-25T18:46:10.358925Z "Boiti, Chiara" https://zbmath.org/authors/?q=ai:boiti.chiara "Jornet, David" https://zbmath.org/authors/?q=ai:jornet.david "Oliaro, Alessandro" https://zbmath.org/authors/?q=ai:oliaro.alessandro "Schindl, Gerhard" https://zbmath.org/authors/?q=ai:schindl.gerhard This article completes the study begun by the first three authors in their paper [in: Advances in microlocal and time-frequency analysis. Contributions of the conference on microlocal and time-frequency analysis 2018, MLTFA18, in honor of Prof. Luigi Rodino on the occasion of his 70th birthday, Torino, Italy, July 2--6, 2018. Cham: Birkhäuser. 121--129 (2020; Zbl 1457.46052)]. The authors use techniques from time-frequency analysis to show that the space $$S_{\omega}$$ of rapidly decreasing ultradifferentiable functions is nuclear for every weight function $$\omega$$ such that $$\omega(t) = o(t)$$ as $$t$$ goes to infinity. Moreover, they show that for a sequence $$(M_p)$$ satisfying the classical condition $$(M1)$$ of Komatsu, the Beurling space $$S_{(M_p)}$$ when defined with $$L_2$$-norms is nuclear if and only if $$(M_p)$$ satisfies condition $$(M2)'$$ of Komatsu. The present research has been continued and extended by the authors in [Banach J. Math. Anal. 15, No. 1, Paper No. 14, 38 p. (2021; Zbl 1472.46043)]. Related work has been published by \textit{A. Debrouwere} et al. [Proc. Am. Math. Soc. 148, No. 12, 5171--5180 (2020; Zbl 07268384); Collect. Math. 72, No. 1, 203--227 (2021; Zbl 1465.46003)]. Pseudodifferential operators in weighted Hölder-Zygmund spaces of variable smoothness https://zbmath.org/1472.47037 2021-11-25T18:46:10.358925Z "Kryakvin, Vadim" https://zbmath.org/authors/?q=ai:kryakvin.v-d "Rabinovich, Vladimir" https://zbmath.org/authors/?q=ai:rabinovich.vladimir-s Summary: We consider pseudodifferential operators of variable orders acting in Hölder-Zygmund spaces of variable smoothness. We prove the boundedness and compactness of the operators under consideration and study the Fredholm property of pseudodifferential operators with slowly oscillating at infinity symbols in the weighted Hölder-Zygmund spaces of variable smoothness. For the entire collection see [Zbl 1367.47005]. The $$L^p$$-Calderón-Zygmund inequality on non-compact manifolds of positive curvature https://zbmath.org/1472.53051 2021-11-25T18:46:10.358925Z "Marini, Ludovico" https://zbmath.org/authors/?q=ai:marini.ludovico "Veronelli, Giona" https://zbmath.org/authors/?q=ai:veronelli.giona Summary: We construct, for $$p> n$$, a concrete example of a complete non-compact $$n$$-dimensional Riemannian manifold of positive sectional curvature which does not support any $$L^p$$-Calderón-Zygmund inequality: \begin{aligned}\Vert\mathrm{Hess}\varphi\Vert_{L^p}\le C(\Vert\varphi\Vert_{L^p}+\Vert\Delta\varphi\Vert_{L^p}),\qquad\forall\varphi\in C^{\infty}_c(M).\end{aligned} The proof proceeds by local deformations of an initial metric which (locally) Gromov-Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez-Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the $$L^p$$-gradient estimates and the $$L^p$$-Calderón-Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds. The smallest eigenvalue distribution of the Jacobi unitary ensembles https://zbmath.org/1472.60011 2021-11-25T18:46:10.358925Z "Lyu, Shulin" https://zbmath.org/authors/?q=ai:lyu.shulin "Chen, Yang" https://zbmath.org/authors/?q=ai:chen.yang.1 Summary: In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $$x^{\alpha}(1 - x)^{\beta}, x \in [0, 1], \alpha, \beta > -1$$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $$[t, 1]$$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval $$(- a, a), a > 0$$ is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight $$(1 - x^2)^{\beta}, x \in [- 1, 1]$$. Distribution of martingales with bounded square functions https://zbmath.org/1472.60075 2021-11-25T18:46:10.358925Z "Stolyarov, Dmitriy M." https://zbmath.org/authors/?q=ai:stolyarov.dmitry-m "Vasyunin, Vasily" https://zbmath.org/authors/?q=ai:vasyunin.vasily-i "Zatitskiy, Pavel" https://zbmath.org/authors/?q=ai:zatitskii.pavel-b "Zlotnikov, Ilya" https://zbmath.org/authors/?q=ai:zlotnikov.ilya-k Summary: We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and $$p$$-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions. Littlewood-Paley-Stein estimates for non-local Dirichlet forms https://zbmath.org/1472.60132 2021-11-25T18:46:10.358925Z "Li, Huaiqian" https://zbmath.org/authors/?q=ai:li.huaiqian "Wang, Jian" https://zbmath.org/authors/?q=ai:wang.jian.2 Summary: We obtain the boundedness in $$L^p$$ spaces for all $$1<p<\infty$$ of the so-called vertical Littlewood-Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $$1<p\leq 2$$, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the uniformly bounded jumping kernel case to the general case, while for $$2\leq p\leq\infty$$, the Burkholder-Davis-Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those for pure jump symmetric Lévy processes in Euclidean spaces. Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence https://zbmath.org/1472.62031 2021-11-25T18:46:10.358925Z "Allaoui, Soumaya" https://zbmath.org/authors/?q=ai:allaoui.soumaya "Bouzebda, Salim" https://zbmath.org/authors/?q=ai:bouzebda.salim "Chesneau, Christophe" https://zbmath.org/authors/?q=ai:chesneau.christophe "Liu, Jicheng" https://zbmath.org/authors/?q=ai:liu.jicheng Summary: This paper is devoted to the estimation of partial derivatives of multivariate density functions. In this regard, nonparametric linear wavelet-based estimators are introduced, showing their attractive properties from the theoretical point of view. In particular, we prove the strong uniform consistency properties of these estimators, over compact subsets of $$\mathbb{R}^d$$, with the determination of the corresponding convergence rates. Then, we establish the asymptotic normality of these estimators. As a main contribution, we relax some standard dependence conditions; our results hold under a weak dependence condition allowing the consideration of mixing, association, Gaussian sequences and Bernoulli shifts. Wavelet-based Benjamini-Hochberg procedures for multiple testing under dependence https://zbmath.org/1472.62068 2021-11-25T18:46:10.358925Z "Ghosh, Debashis" https://zbmath.org/authors/?q=ai:ghosh.debashis.1|ghosh.debashis Summary: Multiple comparisons methodology has experienced a resurgence of interest due to the increase in high-dimensional datasets generated from various biological, medical and scientific fields. An outstanding problem in this area is how to perform testing in the presence of dependence between the $$p$$-values. We propose a novel approach to this problem based on a spacings-based representation of the Benjamini-Hochberg procedure. The representation leads to a new application of the wavelet transform to effectively decorrelate $$p$$-values. Theoretical justification for the procedure is shown. The power gains of the proposed methodology relative to existing procedures is demonstrated using both simulated and real datasets. Multifractal formalisms for multivariate analysis https://zbmath.org/1472.62073 2021-11-25T18:46:10.358925Z "Jaffard, Stéphane" https://zbmath.org/authors/?q=ai:jaffard.stephane "Seuret, Stéphane" https://zbmath.org/authors/?q=ai:seuret.stephane "Wendt, Herwig" https://zbmath.org/authors/?q=ai:wendt.herwig "Leonarduzzi, Roberto" https://zbmath.org/authors/?q=ai:leonarduzzi.roberto "Abry, Patrice" https://zbmath.org/authors/?q=ai:abry.patrice Summary: Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties. Deterministic sparse sublinear FFT with improved numerical stability https://zbmath.org/1472.65173 2021-11-25T18:46:10.358925Z "Plonka, Gerlind" https://zbmath.org/authors/?q=ai:plonka.gerlind "von Wulffen, Therese" https://zbmath.org/authors/?q=ai:von-wulffen.therese Summary: In this paper we extend the deterministic sublinear FFT algorithm in [\textit{G. Plonka} et al., Numer. Algorithms 78, No. 1, 133--159 (2018; Zbl 06871983)] for fast reconstruction of $$M$$-sparse vectors $$\mathbf{x}$$ of length $$N= 2^J$$, where we assume that all components of the discrete Fourier transform $$\hat{\mathbf{x}}=\mathbf{F}_N\mathbf{x}$$ are available. The sparsity of $$\mathbf{x}$$ needs not to be known a priori, but is determined by the algorithm. If the sparsity $$M$$ is larger than $$2^{J/2}$$, then the algorithm turns into a usual FFT algorithm with runtime $$\mathcal{O}(N\log N)$$. For $$M^2<N$$, the runtime of the algorithm is $$\mathcal{O}(M^2\log N)$$. The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$. Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems https://zbmath.org/1472.82002 2021-11-25T18:46:10.358925Z "Lyu, Shulin" https://zbmath.org/authors/?q=ai:lyu.shulin "Chen, Yang" https://zbmath.org/authors/?q=ai:chen.yang.1 Summary: We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of \textit{C. Min} and \textit{Y. Chen} [Math. Methods Appl. Sci. 42, No. 1, 301--321 (2019; Zbl 1409.33018)] where a second-order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in [\textit{X.-B. Wu} and \textit{S.-X. Xu}, Nonlinearity 34, No. 4, 2070--2115 (2021; Zbl 1470.34238)] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as $$n \rightarrow \infty$$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second-order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system. A nonlocal Weickert type PDE applied to multi-frame super-resolution https://zbmath.org/1472.94002 2021-11-25T18:46:10.358925Z "Ait Bella, Fatimzehrae" https://zbmath.org/authors/?q=ai:ait-bella.fatimzehrae "Hadri, Aissam" https://zbmath.org/authors/?q=ai:hadri.aissam "Hakim, Abdelilah" https://zbmath.org/authors/?q=ai:hakim.abdelilah "Laghrib, Amine" https://zbmath.org/authors/?q=ai:laghrib.amine Summary: In this paper, we propose a nonlocal Weickert type PDE for the multiframe super-resolution task. The proposed PDE can not only preserve singularities and edges while smoothing, but also can keep safe the texture much better. This PDE is based on the nonlocal setting of the anisotropic diffusion behavior by constructing a nonlocal term of Weickert type, which is known by its coherence enhancing diffusion tensor properties. A mathematical study concerning the well-posedness of the nonlocal PDE is also investigated with an appropriate choice of the functional space. This PDE has demonstrated its efficiency by combining the diffusion process of Perona-Malik in the flat regions and the anisotropic diffusion of the Weickert model near strong edges, as well as the ability of the non-local term to preserve the texture. The elaborated experimental results give a great insight into the effectiveness of the proposed nonlocal PDE compared to some PDEs, visually and quantitatively. Image encryption using adaptive multiband signal decomposition https://zbmath.org/1472.94003 2021-11-25T18:46:10.358925Z "Alkishriwo, Osama A. S." https://zbmath.org/authors/?q=ai:alkishriwo.osama-a-s Summary: Due to the rapid growth of multimedia transmission over the internet, the challenges of image security have become an important research topic. In this paper, an Adaptive Multiband Signal Decomposition (AMSD) is proposed and its application for image encryption is explored. Like the conventional multiband wavelet transform, the AMSD can decompose the original image into multiband subimages. The perfect reconstruction of the original image from the decomposed multibands is achieved. In addition, a novel image encryption algorithm based on the adaptive multiband image decomposition with three dimensional discrete chaotic maps is developed and its performance is evaluated using common security analysis methods. Simulation results show that the proposed encryption algorithm has great degree of security and can resist various typical attacks. Shearlets as feature extractor for semantic edge detection: the model-based and data-driven realm https://zbmath.org/1472.94004 2021-11-25T18:46:10.358925Z "Andrade-Loarca, Héctor" https://zbmath.org/authors/?q=ai:andrade-loarca.hector "Kutyniok, Gitta" https://zbmath.org/authors/?q=ai:kutyniok.gitta "Öktem, Ozan" https://zbmath.org/authors/?q=ai:oktem.ozan Summary: Semantic edge detection has recently gained a lot of attention as an image-processing task, mainly because of its wide range of real-world applications. This is based on the fact that edges in images contain most of the semantic information. Semantic edge detection involves two tasks, namely pure edge detection and edge classification. Those are in fact fundamentally distinct in terms of the level of abstraction that each task requires. This fact is known as the distracted supervision paradox and limits the possible performance of a supervised model in semantic edge detection. In this work, we will present a novel hybrid method that is based on a combination of the model-based concept of shearlets, which provides probably optimally sparse approximations of a model class of images, and the data-driven method of a suitably designed convolutional neural network. We show that it avoids the distracted supervision paradox and achieves high performance in semantic edge detection. In addition, our approach requires significantly fewer parameters than a pure data-driven approach. Finally, we present several applications such as tomographic reconstruction and show that our approach significantly outperforms former methods, thereby also indicating the value of such hybrid methods for biomedical imaging. Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling https://zbmath.org/1472.94020 2021-11-25T18:46:10.358925Z "Adcock, Ben" https://zbmath.org/authors/?q=ai:adcock.ben "Antun, Vegard" https://zbmath.org/authors/?q=ai:antun.vegard "Hansen, Anders C." https://zbmath.org/authors/?q=ai:hansen.anders-c Summary: Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modeled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted $$\ell^1$$-regularizer we derive measurement conditions that are sharp up to log factors, in the sense that they agree with the best known measurement conditions for oracle estimators in which the support is known a priori. These guarantees also apply in finite dimensions, and improve existing results for unweighted $$\ell^1$$-regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem. Truncation error estimates for two-dimensional sampling series associated with the linear canonical transform https://zbmath.org/1472.94041 2021-11-25T18:46:10.358925Z "Huo, Haiye" https://zbmath.org/authors/?q=ai:huo.haiye Summary: The linear canonical transform (LCT) provides a more general framework for many well-known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi-dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two-dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples. Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences https://zbmath.org/1472.94043 2021-11-25T18:46:10.358925Z "Karnik, Santhosh" https://zbmath.org/authors/?q=ai:karnik.santhosh "Romberg, Justin" https://zbmath.org/authors/?q=ai:romberg.justin-k "Davenport, Mark A." https://zbmath.org/authors/?q=ai:davenport.mark-a Summary: The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in $$\ell_2(\mathbb{Z})$$ which are strictly bandlimited to a frequency band $$[-W,W]$$ and maximally concentrated in a time interval $$\{0,\dots,N-1\}$$. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in $$\mathbb{C}^N$$ whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band $$[-W,W]$$. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior -- slightly fewer than $$2NW$$ eigenvalues are very close to 1, slightly fewer than $$N-2NW$$ eigenvalues are very close to 0, and very few eigenvalues are not near 1 or 0. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near 0 or 1. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between $$\varepsilon$$ and $$1-\epsilon$$. Also, we obtain bounds detailing how close the first $$\approx 2 N W$$ eigenvalues are to 1 and how close the last $$\approx N-2NW$$ eigenvalues are to 0. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between $$\varepsilon$$ and $$1-\epsilon$$.