Recent zbMATH articles in MSC 42https://zbmath.org/atom/cc/422022-11-17T18:59:28.764376ZWerkzeugRecursively enumerable sets and Fourier serieshttps://zbmath.org/1496.031762022-11-17T18:59:28.764376Z"Scarpellini, Bruno"https://zbmath.org/authors/?q=ai:scarpellini.brunoFor the entire collection see [Zbl 1496.00078].Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domainshttps://zbmath.org/1496.220052022-11-17T18:59:28.764376Z"Caspers, Martijn"https://zbmath.org/authors/?q=ai:caspers.martijn"van Velthoven, Jordy Timo"https://zbmath.org/authors/?q=ai:van-velthoven.jordy-timoThe present paper studies certain estimates which improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for \(G=\mathrm{PSU}(1,1)\). More precisely, let \(\pi_\alpha\) be a holomorphic discrete series representation of a connected semisimple Lie group \(G\) with finite center, acting on a weighted Bergman space \(A^2_\alpha (\Omega)\) on a bounded symmetric domain \(\Omega\), of formal dimension \(d_{\pi_\alpha}\). The authors show that if the Bergman kernel \(k _Z^{(\alpha)}\) is a cyclic vector for the restriction \(\pi_\alpha|_{\Gamma}\) to a lattice \(\Gamma\le G\), then \(\mathrm{vol}(G/\Gamma)d_{\pi_\alpha}\le|\Gamma_Z|^{-1}\).
Reviewer: Andreas Arvanitoyeorgos (Patras)Absolute continuity of non-homogeneous self-similar measureshttps://zbmath.org/1496.280042022-11-17T18:59:28.764376Z"Saglietti, Santiago"https://zbmath.org/authors/?q=ai:saglietti.santiago"Shmerkin, Pablo"https://zbmath.org/authors/?q=ai:shmerkin.pablo-s"Solomyak, Boris"https://zbmath.org/authors/?q=ai:solomyak.borisSummary: We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S.-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.Fractional Fourier transform to stability analysis of fractional difffferential equations with Prabhakar derivativeshttps://zbmath.org/1496.340132022-11-17T18:59:28.764376Z"Deepa, S."https://zbmath.org/authors/?q=ai:deepa.s-n"Ganesh, A."https://zbmath.org/authors/?q=ai:ganesh.anumanthappa"Ibrahimov, V."https://zbmath.org/authors/?q=ai:ibrahimov.vagif-r"Santra, S. S."https://zbmath.org/authors/?q=ai:santra.shyam-sundar"Govindan, V."https://zbmath.org/authors/?q=ai:govindan.vediyappan"Khedher, K. M."https://zbmath.org/authors/?q=ai:khedher.khaled-mohamed"Noeiaghdam, S."https://zbmath.org/authors/?q=ai:noeiaghdam.samadSummary: In this paper, the authors introduce the Prabhakar derivative associated with the generalised Mittag-Leffler function. Some properties of the Prabhakar integrals, Prabhakar derivatives and some of their extensions, like fractional Fourier transform of Prabhakar integrals and fractional Fourier transform of Prabhakar derivatives are introduced. This note aims to study the Mittag-Leffler-Hyers-Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. Furthermore, we give a brief definition of the Mittag-Leffler-Hyers-Ulam problem and a method for solving fractional differential equations using the fractional Fourier transform. We show that the fractional differential equations are Mittag-Leffler-Hyers-Ulam stable in the sense of Prabhakar derivatives.Stepanov-like pseudo almost periodic solutions of class \(r\) in \(\alpha \)-norm under the light of measure theoryhttps://zbmath.org/1496.341142022-11-17T18:59:28.764376Z"Zabsonre, Issa"https://zbmath.org/authors/?q=ai:zabsonre.issa"Nsangou, Abdel Hamid Gamal"https://zbmath.org/authors/?q=ai:nsangou.abdel-hamid-gamal"Kpoumiè, Moussa El-Khalil"https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil"Mboutngam, Salifou"https://zbmath.org/authors/?q=ai:mboutngam.salifouSummary: The aim of this work is to present some interesting results on weighted ergodic functions. We also study the existence and uniqueness of \((\mu,\nu)\)-weighted Stepanov-like pseudo almost periodic solutions class \(r\) for some partial differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and his co-authors.Homogenization of Steklov eigenvalues with rapidly oscillating weightshttps://zbmath.org/1496.350482022-11-17T18:59:28.764376Z"Salort, Ariel M."https://zbmath.org/authors/?q=ai:salort.ariel-martinIn this paper, the author studies the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions:
\[
\begin{cases} -\Delta_pu + |u|^{p-2}u = 0 & \mbox{in}\;\; \Omega\,, \\ |\!\bigtriangledown\! u|^{p-2} \frac{\partial u}{\partial \nu} = \lambda \varrho |u|^{p-2} u & \mbox{on}\;\; \partial \Omega\,,\end{cases} \eqno{(P_\varrho)}
\]
where \(\Omega\subset\mathbb{R}^3\) is an open convex bounded domain with Lipschitz boundary \(\partial\Omega\), \(\Delta_p u :=\operatorname{div}(|\!\bigtriangledown\! u|^{p-2}\bigtriangledown\! u)\) with \(1 < p \), and \(\varrho : \partial \Omega \to \mathbb{R} \) is a function such that, for some fixed constants \(\varrho_{\pm} \),
\[
0 < \varrho_- \le \varrho(x) \le \varrho_+ < \infty\,,\;\; x \in \partial \Omega . \eqno{(1)}
\]
A sequence of functions \(\{\varrho_\varepsilon\}_{\varepsilon>0}\) satisfying (1) is given and the author studies the convergence of eigenvalues of \((P_{\varrho_\varepsilon})\) as \(\varepsilon \to 0\) to the limit problem \((P_{\varrho_0}\) ), where these two problems are defined, respectively as
\[
\begin{cases} -\Delta_pu_\varepsilon + |u_\varepsilon|^{p-2}u_\varepsilon = 0 & \mbox{in}\; \Omega, \cr |\!\bigtriangledown\! u_\varepsilon|^{p-2} \frac{\partial u_\varepsilon}{\partial \nu} = \lambda \varrho_\varepsilon |u_\varepsilon|^{p-2} u_\varepsilon & \mbox{on}\; \partial \Omega, \cr \end{cases} \begin{cases} -\Delta_pu_0 + |u_0|^{p-2}u_0 = 0 & \mbox{in}\; \Omega, \cr |\!\bigtriangledown\! u_0|^{p-2} \frac{\partial u_0}{\partial \nu} = \lambda \varrho_0 |u_0|^{p-2} u_0 & \mbox{on}\; \partial \Omega, \cr \end{cases} \eqno{(1.2)}
\]
The main goal of this paper is to study the behavior of the (variational) eigenvalues to \((P_{\varrho_\varepsilon} )\) as \(\varepsilon \to 0\). When no periodicity assumptions on the family \(\{\varrho_{\varepsilon}\}_{\varepsilon>0}\) are made, in Theorem 5.1 the author proves that
\[
\lim_{\varepsilon\to 0}\lambda_{k,\varepsilon} = \lambda_{k,0}
\]
where \(\lambda_{k,\varepsilon}\) and \(\lambda_{k,0 }\) denote the \(k\)-th (\(k \in \mathbb{N}\)) variational eigenvalue of problems (1.2), respectively.
In the case of periodic homogenization, the author obtains explicit estimates of the convergence rate for the first two eigenvalues.
As an application of Theorem 5.1 the author provides some estimates on the first nontrivial curve of the Dancer-Fučík spectrum with Steklov boundary condition.
These results are still true when changing the \(p\)-Laplacian operator with a general quasilinear operator of the form \(\operatorname{div}(|A(x)\!\bigtriangledown\! u \cdot \bigtriangledown u|^{\frac{p-2}{2}} A(x)\bigtriangledown\! u)\), being \(A\) a uniformly elliptic and symmetric matrix.
Reviewer: Petr Tomiczek (Plzeň)Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaceshttps://zbmath.org/1496.351692022-11-17T18:59:28.764376Z"Bhimani, Divyang G."https://zbmath.org/authors/?q=ai:bhimani.divyang-gSummary: We study the Cauchy problems for the Klein-Gordon (HNLKG), wave (HNLW), and Schrodinger (HNLS) equations with cubic convolution (of Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrodinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. As a consequence, we get local and global well-posedness and scattering in larger than usual \(L^p\)-Sobolev spaces.The Dirichlet problem for elliptic operators having a BMO anti-symmetric parthttps://zbmath.org/1496.351842022-11-17T18:59:28.764376Z"Hofmann, Steve"https://zbmath.org/authors/?q=ai:hofmann.steve"Li, Linhan"https://zbmath.org/authors/?q=ai:li.linhan"Mayboroda, Svitlana"https://zbmath.org/authors/?q=ai:mayboroda.svitlana"Pipher, Jill"https://zbmath.org/authors/?q=ai:pipher.jill-cAuthors' abstract: The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation \(\text{div}(A\nabla u)=0\) in the upper half-space \((x,t)\in{\mathbb{R}}^{n+1}_+\) is uniquely solvable when \(n\geq 2\) and the boundary data is in \(L^p({\mathbb{R}}^n,dx)\) for some \(p\in (1,\infty).\) This result is equivalent to saying that the elliptic measure associated to \(L\) belongs to the \(A_\infty\) class with respect to the Lebesgue measure \(dx,\) a quantitative version of absolute continuity.
Reviewer: Dian K. Palagachev (Bari)Pointwise convergence for the elastic wave equationhttps://zbmath.org/1496.352462022-11-17T18:59:28.764376Z"Cho, Chu-Hee"https://zbmath.org/authors/?q=ai:cho.chu-hee"Kim, Seongyeon"https://zbmath.org/authors/?q=ai:kim.seongyeon"Kwon, Yehyun"https://zbmath.org/authors/?q=ai:kwon.yehyun"Seo, Ihyeok"https://zbmath.org/authors/?q=ai:seo.ihyeokSummary: We study pointwise convergence of the solution to the elastic wave equation to the initial data which lies in the Sobolev spaces. We prove that the solution converges along every line to the initial data almost everywhere whenever the initial regularity is greater than one half. We show this is almost optimal.Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimizationhttps://zbmath.org/1496.410072022-11-17T18:59:28.764376Z"Tanaka, Ken'ichiro"https://zbmath.org/authors/?q=ai:tanaka.kenichiro"Sugihara, Masaaki"https://zbmath.org/authors/?q=ai:sugihara.masaakiSummary: We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then we adopt the minimum worst error of the \(n\)-point approximation formulas in each space for characterizing the optimal sampling points for the approximation. In order to obtain approximately optimal sampling points we consider minimization of a discrete energy related to the minimum worst error. Consequently, we obtain an approximation formula and its theoretical error estimate in each space. In addition, from some numerical experiments, we observe that the formula generated by the proposed method outperforms the corresponding formula derived with sinc approximation, which is near optimal in each space.Approximation by linear combinations of translates of a single functionhttps://zbmath.org/1496.410122022-11-17T18:59:28.764376Z"Dũng, Dinh"https://zbmath.org/authors/?q=ai:dinh-dung."Huy, Vu Nhat"https://zbmath.org/authors/?q=ai:huy.vu-nhatSummary: We study approximation of periodic functions by arbitrary linear combinations of n translates of a single function. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the \(L^p\)-approximation convergence rate by these methods, when \(n\rightarrow\infty\), for \(1\leq p\leq\infty\). We also generalize these results to classes of multivariate functions defined as the convolution with the tensor product of a single function. In the case \(p=2\), for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of \(n\) translates of arbitrary function.Approximation by quasi-interpolation operators and Smolyak's algorithmhttps://zbmath.org/1496.420012022-11-17T18:59:28.764376Z"Kolomoitsev, Yurii"https://zbmath.org/authors/?q=ai:kolomoitsev.yurii-sSummary: We study approximation of multivariate periodic functions from Besov and Triebel-Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the \(L_q\)-norm for functions from the Besov spaces \(\text{B}_{p,\theta}^s(\mathbb{T}^d)\) and the Triebel-Lizorkin spaces \(\text{F}_{p,\theta}^s(\mathbb{T}^d)\) for all \(s>0\) and admissible \(1 \leq p\), \(\theta\leq\infty\) as well as provide analogues of the Littlewood-Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.Functions with general monotone Fourier coefficientshttps://zbmath.org/1496.420022022-11-17T18:59:28.764376Z"Belov, Aleksandr S."https://zbmath.org/authors/?q=ai:belov.aleksandr-s"Dyachenko, Mikhail I."https://zbmath.org/authors/?q=ai:dyachenko.mikhail-ivanovich"Tikhonov, Sergei Yu."https://zbmath.org/authors/?q=ai:tikhonov.sergey-yuModulation spaces, multipliers associated with the special affine Fourier transformhttps://zbmath.org/1496.420032022-11-17T18:59:28.764376Z"Biswas, M. H. A."https://zbmath.org/authors/?q=ai:biswas.md-haider-ali|biswas.md-hasan-ali"Feichtinger, H. G."https://zbmath.org/authors/?q=ai:feichtinger.hans-georg"Ramakrishnan, R."https://zbmath.org/authors/?q=ai:ramakrishnan.ram-t-s|ramakrishnan.raghu|ramakrishnan.ramya|ramakrishnan.rajasekhar|ramakrishnan.ravi|ramakrishnan.ramkumar|ramakrishnan.rishiSummary: We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space \(\mathcal{M}^{r, s}_A\) in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with \(A\)-translation, then it is a \(A\)-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.Lipschitz and Fourier type conditions with moduli of continuity in rank \(1\) symmetric spaceshttps://zbmath.org/1496.420042022-11-17T18:59:28.764376Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Restrepo, Joel E."https://zbmath.org/authors/?q=ai:restrepo.joel-esteban"Suragan, Durvudkhan"https://zbmath.org/authors/?q=ai:suragan.durvudkhanSummary: Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an \(L^2\) function, in the setting of Riemannian symmetric spaces of rank \(1\) whose growth depends on a \(k\)th-order modulus of continuity.A noncommutative approach to the graphon Fourier transformhttps://zbmath.org/1496.420052022-11-17T18:59:28.764376Z"Ghandehari, Mahya"https://zbmath.org/authors/?q=ai:ghandehari.mahya"Janssen, Jeannette"https://zbmath.org/authors/?q=ai:janssen.jeannette-c-m"Kalyaniwalla, Nauzer"https://zbmath.org/authors/?q=ai:kalyaniwalla.nauzerSummary: Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon. Theoretically, graphons are limit objects of converging sequences of graphs. Our work extends previous research proposing a common scheme for signal analysis of graphs that are similar in structure to a graphon. We extend a previous definition of graphon Fourier transform, and show that the graph Fourier transforms of graphs in a converging graph sequence converge to the graphon Fourier transform of the limiting graphon. We then apply this convergence result to signal processing on Cayley graphons. We show that Fourier analysis of the underlying group enables the construction of a suitable eigen-decomposition for the graphon, which can be used as a common framework for signal processing on graphs converging to the graphon.Time-frequency transforms with Poisson kernel modulationhttps://zbmath.org/1496.420062022-11-17T18:59:28.764376Z"Zhang, Yiqiao"https://zbmath.org/authors/?q=ai:zhang.yiqiao"Chen, Qiuhui"https://zbmath.org/authors/?q=ai:chen.qiuhuiAP-frames and stationary random processeshttps://zbmath.org/1496.420072022-11-17T18:59:28.764376Z"Centeno, Hernán D."https://zbmath.org/authors/?q=ai:centeno.hernan-d"Medina, Juan M."https://zbmath.org/authors/?q=ai:medina.juan-miguelSummary: It is known that, in general, an AP-frame is an \(L^2 (\mathbb{R})\)-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system \(\{ g(t - k) e^{il(t-k)}, l \in \mathbb{L} = \omega_0 \mathbb{Z}, k \in \mathbb{K} = t_0 \mathbb{Z}\}\) to be an \(L^2 (\mathbb{R})\)-Frame in terms of Gaussian stationary random processes. In addition, if \(X = (X(t))_{t \in \mathbb{R}}\) is a wide sense stationary random process, we study density conditions for the associated stationary sequences \(\{\langle X, g_{k, l} \rangle, l \in \mathbb{L}, k \in \mathbb{K}\}\).Completeness conditions of systems of Bessel functions in weighted \(L^2\)-spaces in terms of entire functionshttps://zbmath.org/1496.420082022-11-17T18:59:28.764376Z"Khats', Ruslan"https://zbmath.org/authors/?q=ai:khats.r-vSummary: Let \(J_\nu\) be the Bessel function of the first kind of index \(\nu\geq 1/2\), \(p\in\mathbb{R}\) and \((\rho_k)_{k\in\mathbb{N}}\) be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system \(\left\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k\in\mathbb{N}\right\}\) in the weighted space \(L^2((0;1); x^{2p} dx)\) are found in terms of an entire function with the set of zeros coinciding with the sequence \((\rho_k)_{k\in\mathbb{N}}\).Multi-dimensional \(c\)-almost periodic type functions and applicationshttps://zbmath.org/1496.420092022-11-17T18:59:28.764376Z"Kostic, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this article, we analyze multi-dimensional Bohr \((\mathcal{B}, c)\)-almost periodic type functions. The main structural characterizations for the introduced classes of Bohr \((\mathcal{B}, c)\)-almost periodic type functions are established. Several applications of our abstract theoretical results to the abstract Volterra integro-differential equations in Banach spaces are provided, as well.Positive definiteness and infinite divisibility of certain functions of hyperbolic cosine functionhttps://zbmath.org/1496.420102022-11-17T18:59:28.764376Z"Kosaki, Hideki"https://zbmath.org/authors/?q=ai:kosaki.hidekiLet \(\alpha \geq 0\), \(t>-1\) and \(f_{\alpha },\) \(g_{\alpha }\) two real functions defined by
\[
f_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (3x)+t\cosh x}
\]
and
\[
g_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (2x)+t\cosh x}.
\]
The author investigates infinite divisibility and positive definiteness of the functions \(f_{\alpha }\) and of \(g_{\alpha }\). Furthermore, he uses the positive definiteness criterion to study certain norm comparison results for operator means.
Reviewer: Elhadj Dahia (Bou Saâda)Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier serieshttps://zbmath.org/1496.420112022-11-17T18:59:28.764376Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferencThe author proved that the triangular Cesàro means (of positive order) of bivariable functions \(f\in L^{p}(T^{2})\) with \(1\leq p<\infty \) converge to \(f\) at each strong \((1,\omega )\)-Lebesgue point. This generalizes the well-known classical Lebesgue's theorem.
Reviewer: Włodzimierz Łenski (Poznań)Quaternionic linear canonical wave packet transformhttps://zbmath.org/1496.420122022-11-17T18:59:28.764376Z"Bhat, Younis Ahmad"https://zbmath.org/authors/?q=ai:bhat.younis-ahmad"Sheikh, N. A."https://zbmath.org/authors/?q=ai:sheikh.neyaz-ahmed|sheikh.nadeem-ahmad|sheikh.neya-ahmad|sheikh.neyaz-ahmadThe wave packet transform is defined as the Fourier transform of a signal windowed with a given wavelet. This transform is later generalized by \textit{A. Prasad} and \textit{M. Kundu} [Integral Transforms Spec. Funct. 32, No. 11, 893--911 (2021; Zbl 1479.42019)] as linear canonical wave packet transform.
In this paper, the authors extend the concept of linear canonical wave packet transform to quaternion-valued signals. Essential properties (including uncertainty principles) of this quaternion linear canonical wave packet transform (QLCWPT) are presented. Applications of QLCWPT are not given.
Reviewer: Manfred Tasche (Rostock)Inequalities for Dunkl-Riesz transforms and Dunkl gradient with radial piecewise power weightshttps://zbmath.org/1496.420132022-11-17T18:59:28.764376Z"Ivanov, Valeriĭ Ivanovich"https://zbmath.org/authors/?q=ai:ivanov.valerii-ivanovichSummary: A beautiful and meaningful harmonic analysis has been constructed on the Euclidean space \(\mathbb{R}^d\) with Dunkl weight. The classical Fourier analysis on \(\mathbb{R}^d\) corresponds to the weightless case. The Dunkl-Riesz potential and the Dunkl-Riesz transforms play an important role in the Dunkl harmonic analysis. In particular, they allow one to prove the Sobolev type inequalities for the Dunkl gradient. Earlier we proved \((L^q,L^p)\)-inequalities for the Dunkl-Riesz potential with two radial piecewise power weights. For the Dunkl-Riesz transforms, we proved \(L^p\)-inequality with one radial power weight and, as a consequence, we obtained \((L^q,L^p)\)-inequalities for the Dunkl gradient with two radial power weights. In this paper, these results for the Dunkl-Riesz transforms and the Dunkl gradient for radial power weights are generalized to the case of radial piecewise power weights.Fourier transform of anisotropic mixed-norm Hardy spaces with applications to Hardy-Littlewood inequalitieshttps://zbmath.org/1496.420142022-11-17T18:59:28.764376Z"Liu, Jun"https://zbmath.org/authors/?q=ai:liu.jun.4"Lu, Yaqian"https://zbmath.org/authors/?q=ai:lu.yaqian"Zhang, Mingdong"https://zbmath.org/authors/?q=ai:zhang.mingdongSummary: Let \(\vec{p}\in(0,1]^n\) be an \(n\)-dimensional vector and \(A\) a dilation. Let \(H_A^{\vec{p}}(\mathbb{R}^n)\) denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of \(H_A^{\vec{p}}(\mathbb{R}^n)\) and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of \(f\in H_A^{\vec{p}}(\mathbb{R}^n)\) coincides with a continuous function \(F\) on \(\mathbb{R}^n\) in the sense of tempered distributions. Moreover, the function \(F\) can be controlled pointwisely by the product of the Hardy space norm of \(f\) and a step function with respect to the transpose matrix of \(A\). As applications, the authors obtain a higher order of convergence for the function \(F\) at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of \(H_A^{\vec{p}}(\mathbb{R}^n)\).Multi-dimensional linear canonical transform with applications to sampling and multiplicative filteringhttps://zbmath.org/1496.420152022-11-17T18:59:28.764376Z"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmad"Tantary, Azhar Y."https://zbmath.org/authors/?q=ai:tantary.azhar-yGiven a \(2n\times 2n\) symplectic matrix \(M=\left(\begin{matrix} A & B\\
C & D\end{matrix}\right)\), that is, \(M^T J M=J\) where \(J=\left(\begin{matrix} 0 & I_n\\
-I_n & 0\end{matrix}\right)\), the (multidimensional) linear canonical transform (LCT) \(\mathcal{L}_M\) of \(f\in L^2(\mathbb{R}^n)\) is
\[
\mathcal{L}_M[f](w)=\int_{\mathbb{R}^n} f(x) K_M(x,w)\, dx
\]
where (with \(\Omega(B,n)=(2\pi)^{-n/2} |\det{B}|^{-1/2}\) and \(|\det{B}|\neq 0\))
\[
K_M(x,w)=\Omega(B,n)\exp\left\{\frac{i}{2}(w^T DB^{-1}w-2w^TB^{-T}x+x^TB^{-1}Ax)\right\}\, .
\]
The Fourier transform corresponds to \(M=J\). The LCT satisfies the Parseval property \(\langle \mathcal{L}_M[f], \, \mathcal{L}_M[g]\rangle=\langle f,\, g\rangle\).
The authors define a linear canonical convolution by
\[
(f \circledast_M g)(t) =\int_{\mathbb{R}^n} f(x) g(t-x) \exp\left\{\frac{i}{2}(x^T B^{-1} A(x-t) + (x-t)^T B^{-1}A x )\right\}\, dx
\]
and prove (Thm.~2.1) that
\[
\mathcal{L}_M[(f \circledast_M g)](w)=\frac{\exp\left\{-i(w^T DB^{-1}w)/2\right\}}{\Omega(B,n)} \mathcal{L}_M[(f )](w) \mathcal{L}_M[(g )](w)\, .
\]
A corresponding result (Thm.~2.3) is proved for what the authors call the linear canonical correlation, which is the same as the convolution but with \(g\) replaced by \(\bar{g}\) in the corresponding convolution/transform integral (and with \(\mathcal{L}_M[(g )]\) replaced by \(\overline{\mathcal{L}_M[(g )]}\) in the corresponding product formula).
An analogue of the Heisenberg uncertainty inequality is proved (Thm.~3.1): \(\Delta_{f}^2\Delta_{\mathcal{L}_M[(f )]}^2\geq \frac{n^2}{4} |\det{B}|^2\) where \(\Delta_f^2=\frac{\|(x-x_0) f(x)\|^2}{\|f\|^2}\) with \(x_0=\|xf\|^2/\|f\|^2\). The inequality is an identity when \(f\) is a multiple of \(\exp\left(-\frac{ic x^TB^{-1}A x+2|x|^2}{2c}\right)\) for some \(c>0\). A version of Beckner's logarithmic (entropic) uncertainty inequality is also proved (Thm.~3.2), namely
\[
\int \ln |w| |\mathcal{L}_M[f](w)|^2\, dw +\int \ln |x| |f(x)|^2\, dx\geq \left[\frac{\Gamma'(n/4)}{\Gamma(n/4)}+\ln |\det{B}|-\ln\pi\right]\|f\|^2\, .
\]
Perhaps the main result (Thm.~4.1) is a sampling formula analogous to Shannon's sampling theorem in the context of the LCT. One assumes that \(f\in L^2(\mathbb{R}^n)\) is bandlimited in the LCT domain to a region \(\mathcal{D}\) that is optimally contained in a regular polyhedral region denoted by \(\tilde{\mathbb{U}}(2E)\). Then \(f\) can be expressed in terms of its samples \(f(Nk)\) via
\begin{multline*}
f(x)=\frac{\det{N}}{(2\pi)^n|\det{B}|}\exp\left\{-\frac{i}{2}(x^T B^{-1}Ax \right\} \sum_{k\in\mathbb{Z}^n} f(Nk)\\
\times \left[ \exp\left\{-\frac{i}{2}((Nk)^T B^{-1}A(Nk) \right\} \left\{ \int_{\tilde{\mathbb{U}}(2E)}\exp\left\{-i((B^{-1}w)^T)(x-Nk) \right\} dw\right\} \right]
\end{multline*}
provided \(EK=\pi BN^{-T}\) for some non-singular integer matrix \(K\). Here \(\mathbb{U}(N)=\{Nx: x\in [0,1)^n\}\) for a nonsingular \(n\times n\) matrix \(N\), and \(\tilde{\mathbb{U}}(2N)\) is a translation of \(\mathbb{U}(2N)\). An ample discussion of the lattice coset geometry that underlies the result is provided and the result is illustrated for specific choices of \(M\).
The authors assert that the approach here is the first that takes full advantage of the \(n(2n+1)\) dimensionality of the symplectic group.
Reviewer: Joseph Lakey (Las Cruces)The Brascamp-Lieb inequality and its influence on Fourier analysishttps://zbmath.org/1496.420162022-11-17T18:59:28.764376Z"Zhang, Ruixiang"https://zbmath.org/authors/?q=ai:zhang.ruixiangSummary: Brascamp-Lieb inequalities have been important in analysis, mathematical physics and neighboring areas. Recently, these inequalities have had a deep influence on Fourier analysis and, in particular, on Fourier restriction theory. In this chapter we motivate and explain this connection. A lot of our examples are taken from a rapidly developing subarea called ``decoupling''. It is the author's hope that this chapter will be accessible to graduate students in fields broadly related to analysis.
For the entire collection see [Zbl 1491.46003].Erratum to: ``The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces''https://zbmath.org/1496.420172022-11-17T18:59:28.764376Z"Fang, Chenglong"https://zbmath.org/authors/?q=ai:fang.chenglong"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangErratum to the authors' paper [ibid. 52, No. 2, 375--383 (2021; Zbl 1480.42019)].On the sharp-weighted norm estimate for the commutator of Littlewood-Paley operatorshttps://zbmath.org/1496.420182022-11-17T18:59:28.764376Z"Ghorbanalizadeh, Arash"https://zbmath.org/authors/?q=ai:ghorbanalizadeh.arash-m"Hasanvandi, Sajjad"https://zbmath.org/authors/?q=ai:hasanvandi.sajjadSummary: Using a classical Cauchy integral argument, we obtain sharp-weighted inequality for \(m\)th order commutator of a square function with \(b \in \mathrm{BMO}\). Also, the \(A_p-A_{\infty}\) estimate for commutators of square functions is proved.Weighted endpoint estimates for the composition of Calderón-Zygmund operators on spaces of homogeneous typehttps://zbmath.org/1496.420192022-11-17T18:59:28.764376Z"Liu, Dongli"https://zbmath.org/authors/?q=ai:liu.dongli"Zhao, Jiman"https://zbmath.org/authors/?q=ai:zhao.jimanSummary: Let \(T_1\), \(T_2\) be two Calderón-Zygmund operators, by establishing bilinear sparse domination of \(T_1 T_2\), we obtain the weighted endpoint estimate for the composite operator \(T_1 T_2\).On a generalization of the Hörmander conditionhttps://zbmath.org/1496.420202022-11-17T18:59:28.764376Z"Suzuki, Soichiro"https://zbmath.org/authors/?q=ai:suzuki.soichiroIn this article, the author considered a natural generalization of the classical Hörmander condition in the Calderón-Zygmund theory. The author proved that the \(L^1\) mean condition actually coincides with the classical one. Meanwhile, the author introduced a new variant of the Hörmander condition, which is strictly weaker than the classical one but still enough for the \(L^p\) boundedness. Moreover, it still works in the non-doubling setting with a little modification.
Personally speaking, this paper is very interesting and very well written. This paper involves a large number of definitions, notations, and references, which increases its richness. Overall, this article is a nice piece of work.
Reviewer: Feng Liu (Qingdao)The dual spaces of variable anisotropic Hardy-Lorentz spaces and continuity of a class of linear operatorshttps://zbmath.org/1496.420212022-11-17T18:59:28.764376Z"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Wang, Aiting"https://zbmath.org/authors/?q=ai:wang.aitingSummary: In this paper, the authors obtain the continuity of a class of linear operators on variable anisotropic Hardy-Lorentz spaces. In addition, the authors also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type space with variable exponents. This result is still new even when the exponent function \(p(\cdot)\) is \(p\).The circular maximal operator on Heisenberg radial functionshttps://zbmath.org/1496.420222022-11-17T18:59:28.764376Z"Beltran, David"https://zbmath.org/authors/?q=ai:beltran.david"Guo, Shaoming"https://zbmath.org/authors/?q=ai:guo.shaoming"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathan"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreasDenote by \(\mathbb{H}^n\) the Heisenberg group, the set \(\mathbb{R}\times\mathbb{R}^{2n}\) equipped with the following non-commutative group operation: for all \((u,x), (v,y)\in\mathbb{H}^n\),
\[
(u,x)\cdot (v,y):=(u+v+x^TBy,x+y).
\]
Here, \(B:=\begin{pmatrix}0&-bI_n\\
bI_n&0\end{pmatrix}\) for some \(b\neq 0\) (one typically choose \(b=1/2\)). For \(\mu_1\equiv\mu\), the normalized surface measure on \(\{0\}\times\mathbb{S}^{2n-1}\), let \(\mu_t\) denote its dilation supported on \(t\mathbb{S}^{2n-1}\). For a function \(f:\mathbb{H}^n\rightarrow\mathbb{C}\), one may formally define its spherical means as \[ f\ast\mu_t(u,x):=\int_{\mathbb{S}^{2n-1}}\!f(u-tx^TBy,x-ty)\,d\mu(y) \] and its spherical maximal function as
\[
Mf(u,x):=\sup_{t>0}|f\ast\mu_t(u,x)|.
\]
In this paper, the authors complement known \(L^p\)-boundedness results for \(M\) on \(\mathbb{H}^n\), \(n\geq{2}\), by initiating the study of the case \(n=1\), where currently nothing is known for any \(p<\infty\). For \(2<p\leq\infty\), they show the existence of a constant \(C_p\), depending only on \(p\), such that
\[
\|Mf\|_{L^p(\mathbb{H}^1)}\leq C_p\|f\|_{L^p(\mathbb{H}^1)}
\]
for all \(\mathbb{H}\)-radial functions \(f\) on \(\mathbb{H}^1\). A function \(f:\mathbb{H}^1\rightarrow\mathbb{C}\) is said to be \(\mathbb{H}\)-radial if \(f(u,Rx)=f(u,x)\) for all \((u,x)\in\mathbb{H}^1\) and all \(R\) belonging to the special orthogonal group, \(SO(2)\). Equivalently, \(f\) is \(\mathbb{H}\)-radial if and only if there exists some function \(f_0:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{C}\) such that \(f(u,x)=f_0(u,|x|)\) for all \((u,x)\in\mathbb{H}^1\).
The authors accomplish this by reducing the problem to studying the boundedness of a maximal function given by \(\sup_{t>0} |A_tf|\), where \(\{A_t\}\) are non-convolution averaging operators on \(\mathbb{R}^2\). While the reduction is not difficult, the associated curve distribution has vanishing rotational and cinematic curvatures, precluding the straightforward application of the standard techniques used to study the Euclidean spherical maximal function. A significant portion of this paper is spent overcoming these challenges, along the way performing an \(L^2\) analysis of two-parameter oscillatory integrals with two-sided fold singularities.
The appendices contain, among other things, a discussion of the use of repeated integration by parts often seen when studying oscillatory integrals.
Reviewer: Ryan Gibara (Cincinnati)On boundedness of maximal operators associated with hypersurfaceshttps://zbmath.org/1496.420232022-11-17T18:59:28.764376Z"Ikromov, I. A."https://zbmath.org/authors/?q=ai:ikromov.isroil-a"Usmanov, S. E."https://zbmath.org/authors/?q=ai:usmanov.s-eSummary: In this paper, we obtain a criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also, we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the Varchenko height of the hypersurface is greater than two. We obtain the exact value of the boundedness index for degenerate smooth hypersurfaces, i.e., for hypersurfaces satisfying the assumptions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer conjecture for arbitrary convex analytic hypersurfaces as well as for smooth degenerate hypersurfaces. Also, we discuss related problems of the theory of oscillatory integrals.Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functionshttps://zbmath.org/1496.420242022-11-17T18:59:28.764376Z"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng"Wu, Huoxiong"https://zbmath.org/authors/?q=ai:wu.huoxiong"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingying"Yabuta, Kôzô"https://zbmath.org/authors/?q=ai:yabuta.kozoThe centered maximal function is denoted \(\mathcal{M}\) and the uncentered maximal function as \(\widetilde{\mathcal{M}}\); I will give the results for the uncentered maximal functions and let the interested reader look at the paper for results related to the centered maximal function. Those results may be slightly different.
There is much that is understood about derivatives of the Hardy-Littlewood maximal function, for example \(\widetilde{\mathcal{M}}:W^{1,p}(\mathbb{R}^n) \mapsto W^{1,p}(\mathbb{R}^n), 1< p \leq \infty\). Because of non-reflexivity, the situation at \(p = 1\) is more delicate, with the basic question being whether \(f \mapsto \nabla\widetilde{ \mathcal{M}}f\) maps \(W^{1,1}(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\). \textit{J. M. Aldaz} and \textit{J. Pérez Lázaro} [Trans. Am. Math. Soc. 359, No. 5, 2443--2461 (2007; Zbl 1143.42021)] proved that if \(f\) is of bounded variation, \(\widetilde{\mathcal{M}}f\) is absolutely continuous and
\[
\mbox{Var} ( \widetilde{\mathcal{M}f}) \leq \mbox{Var}(f),
\]
and thus that
\[
|| (\widetilde{\mathcal{M}}f)^{\prime} ||_{L^1(\mathbb{R})} \leq || f||_{L^1(\mathbb{R})} , f \in W^{1,1}(\mathbb{R}),
\]
and the constant is sharp.
\textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] proved one-dimensional results for the fractional maximal operator
\[
\widetilde{\mathcal{M}}_{\alpha} f(x) = \sup_{r, s \geq 0, r + s>0} \frac{1}{(r +s)^{1 - \alpha}} \int_{x - r}^{x +s} |f(y)| \, dy
\]
(Note that \( \widetilde{\mathcal{M}}_{0} = \widetilde{\mathcal{M}}\)). The ultimate result was in [\textit{J. Madrid}, Rev. Mat. Iberoam. 35, No. 7, 2151--2168 (2019; Zbl 1429.42021)] who proved that if \(0 < \alpha <1, q = \frac{1}{1 - \alpha}, f \mapsto (\widetilde{\mathcal{M}}_{\alpha} f)^{\prime} \) is continuous from \(W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\).
One of the authors' results is to extend this to the multilinear case. If \(\vec{f} = (f_1, \ldots, f_m)\), where each \(f_j \in L^1_{\mbox{loc}}(\mathbb{R})\), the uncentered fractional maximal operator is
\[
\widetilde{\mathfrak{M}}_{\alpha} \vec{f}(x) = \sup_{r, s \geq 0, r + s>0} \frac{1}{(r +s)^{m - \alpha}} \prod_{j = 1}^m \int_{x - r}^{x +s} |f_j(y)| \, dy,
\]
and it is known that it sends \(W^{1,p_1}(\mathbb{R}) \times \cdots \times W^{1,p_m}(\mathbb{R})\) to \(W^{1,q}(\mathbb{R})\) provided that \(1 < p_1,\dots, p_m, 0 < \alpha <m, 1/q = 1/p_1 + \cdots + 1/p_m - \alpha\). The question is about the behavior of the endpoint case \(p_1 = p_2 = \cdots = p_m = 1\). Question A is whether the mapping \(\vec{f} \mapsto (\widetilde{\mathcal{\mathcal{M}}}_{\alpha} f)^{\prime} \) bounded and continuous from \(W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\) if \(0 < \alpha <m , q = \frac{1}{m - \alpha}\). Results of \textit{F. Liu} and \textit{H. Wu} [Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and others can be extended to handle \(m \geq 2, 1 \leq \alpha < m\) and in this paper the authors prove the remaining case; if \(0 < \alpha <1\), \(q = \frac{1}{1 - \alpha}\), then \(\vec{f} \mapsto (\widetilde{\mathfrak{M}}_{\alpha} f)^{\prime} \) is bounded and continuous from \(W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})\) to \(L^q(\mathbb{R})\).
Reviewer: Raymond Johnson (Columbia)Hardy-Littlewood maximal operator on variable Lebesgue spaces with respect to a probability measurehttps://zbmath.org/1496.420252022-11-17T18:59:28.764376Z"Moreno, Jorge"https://zbmath.org/authors/?q=ai:moreno.jorge"Pineda, Ebner"https://zbmath.org/authors/?q=ai:pineda.ebner"Rodriguez, Luz"https://zbmath.org/authors/?q=ai:rodriguez.luz"Urbina, Wilfredo O."https://zbmath.org/authors/?q=ai:urbina-romero.wilfredo-oIn this paper, the authors established the strong and weak boundedness of Hardy-Littlewood maximal operators on variable Lebesgue spaces \(L^{p(\cdot)}(\mu)\) with respect to a probability Borel measure \(\mu\) for two conditions of regularity on the exponent function \(p(\cdot)\).
To be more precise, let \(\mu\) be a Radon measure and \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), The Hardy-Littlewood non-centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu f(x)=\sup_{B\ni x}\fint_{B}|f(x)|\mu(dy),
\]
and the Hardy-Littlewood centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu^c f(x)=\sup_{r>0}\fint_{B(x,r)}|f(y)|\mu(dy),
\]
The authors first prove the boundedness with the condition \(\mathcal{P}^0_\mu(\mathbb{R}^d)\).
Theorem 1. Let \(p(\cdot)\in\mathcal{P}^0_\mu(\mathbb{R}^d)\) be continuous with \(p_- > 1\).
(i) There exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
(ii) If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Then, they gave the boundedness with the condition \(\mathcal{P}_\mu(\mathbb{R}^d)\).
Theorem 2. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) with \(p_- > 1\) be such that \(1/{p(\cdot)}\) is continuous. If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Theorem 3. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) be such that \(1/{p(\cdot)}\) is continuous, then there exists \(C>0\) depending on \(p\) such that
\[
\|t\chi_{\{x\in\mathbb{R}^d:M_\mu^c f(x)>t\}}\|_{p(\cdot),\mu}\leq C\|f\|_{p(\cdot),\mu}
\]
for all \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), \(t>0\).
They also extended some properties of this operator to a probability Borel measure \(\mu\), the key to extending these results is using the Besicovitch covering lemma instead of the Calderón Zygmund decomposition.
Reviewer: Qingying Xue (Beijing)Dimension independent atomic decomposition for dyadic martingale \(\mathbb{H}^1\)https://zbmath.org/1496.420262022-11-17T18:59:28.764376Z"Paluszynski, Maciej"https://zbmath.org/authors/?q=ai:paluszynski.maciej"Zienkiewicz, Jacek"https://zbmath.org/authors/?q=ai:zienkiewicz.jacekSummary: We introduce atoms for dyadic atomic \(\mathbb{H}^1\) for which the equivalence between the atomic and maximal function definitions is dimension independent. We give sharp, up to \(\log(d)\) factor, estimates for the \(\mathbb{H}^1\rightarrow L^1\) norm of the special maximal function.A note on the boundedness of iterated commutators of multilinear operatorshttps://zbmath.org/1496.420272022-11-17T18:59:28.764376Z"Wang, Dinghuai"https://zbmath.org/authors/?q=ai:wang.dinghuaiSummary: We show that the symbol function belonging to \(BMO\) space is not a necessary condition for the boundedness of the iterated commutator acting on a product of Lebesgue spaces.Endpoint Sobolev bounds for fractional Hardy-Littlewood maximal operatorshttps://zbmath.org/1496.420282022-11-17T18:59:28.764376Z"Weigt, Julian"https://zbmath.org/authors/?q=ai:weigt.julianThe paper under review is concerned with the regularity properties of the fractional Hardy-Littlewood maximal function on \(\mathbb{R}^d\). Given a parameter \(\alpha \in [0,d)\), let \(M_\alpha\) denote the centered fractional Hardy-Littlewood maximal function defined by
\[
M_\alpha f(x) = \sup_{r>0} \frac{r^\alpha}{|B(x,r)|} \int_{B(x,r)} |f|,
\]
where \(B(x,r)\) denotes the ball of radius \(r\) centered at \(x\in \mathbb{R}^d\). It is shown that for any \(f\) in the Sobolev space \(W^{1,p}(\mathbb{R}^d)\), \(M_\alpha f\) is weakly differentiable and the following inequality holds with \(1/p^\ast = 1/p - \alpha/d\),
\[
|| \nabla M_\alpha f ||_{p^\ast} \leq C_{d,\alpha,p} || \nabla f ||_p,
\]
provided that \(1\leq p < d/\alpha\) and \(0<\alpha<d\). Similar inequality is established for the uncentered fractional Hardy-Littlewood maximal function. For the endpoint case \(p=1\), the inequality had been left open for the range \(0<\alpha<1\). It is noted that, for the usual Hardy-Littlewood maximal function (the case \(\alpha=0\)), the inequality at the endpoint \(p=1\) is still left open except for the one-dimensional case and some partial results in higher dimensions. For proof of the inequality, the author develops a new inequality for a dyadic version of the fractional Hardy-Littlewood maximal function.
Reviewer: Jongchon Kim (Hong Kong)On BMO and Carleson measures on Riemannian manifoldshttps://zbmath.org/1496.420292022-11-17T18:59:28.764376Z"Brazke, Denis"https://zbmath.org/authors/?q=ai:brazke.denis"Schikorra, Armin"https://zbmath.org/authors/?q=ai:schikorra.armin"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannickSummary: Let \(\mathcal{M}\) be a Riemannian \(n\)-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions \(u: \mathcal{M}\to\mathbb{R}\) by a Carleson measure condition of their \(\sigma\)-harmonic extension \(U: \mathcal{M}\times (0,\infty)\to\mathbb{R}\). We make crucial use of a \(T(b)\) theorem proved by \textit{S. Hofmann} et al. [\(L^p\)-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1371.28004)]. As an application, we show that the famous theorem of Coifman-Lions-Meyer-Semmes [\textit{R. Coifman} et al., J. Math. Pures Appl. (9) 72, No. 3, 247--286 (1993; Zbl 0864.42009)] holds in this class of manifolds: Jacobians of \(W^{1,n}\)-maps from \(\mathcal{M}\) to \(\mathbb{R}^n\) can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by \textit{E. Lenzmann} and \textit{A. Schikorra} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 193, Article ID 111375, 37 p. (2020; Zbl 1436.35012)] using only harmonic extensions, integration by parts, and trace space characterizations.Hardy spaces meet harmonic weightshttps://zbmath.org/1496.420302022-11-17T18:59:28.764376Z"Preisner, Marcin"https://zbmath.org/authors/?q=ai:preisner.marcin"Sikora, Adam"https://zbmath.org/authors/?q=ai:sikora.adam-s"Yan, Lixin"https://zbmath.org/authors/?q=ai:yan.lixinSummary: We investigate the Hardy space \(H^1_L\) associated with a self-adjoint operator \(L\) defined in a general setting by \textit{S. Hofmann} et al. [Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1232.42018)]. We assume that there exists an \(L\)-harmonic non-negative function \(h\) such that the semigroup \(\exp (-tL)\), after applying the Doob transform related to \(h\), satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space \(H^1_L\) in terms of a simple atomic decomposition associated with the \(L\)-harmonic function \(h\). Our approach also yields a natural characterisation of the \(BMO\)-type space corresponding to the operator \(L\) and dual to \(H^1_L\) in the same circumstances.
The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in \({\mathbb{R}^n} \), Schrödinger operators with certain potentials, and Bessel operators.Finite shift-invariant subspaces of periodic functions: characterization, approximation, and applicationshttps://zbmath.org/1496.420312022-11-17T18:59:28.764376Z"Atreas, Nikolaos"https://zbmath.org/authors/?q=ai:atreas.nikolaos-dSummary: We discuss approximations of square integrable periodic functions by their projections in finite shift-invariant subspaces and highlight the role of principal shift invariance. We also show how we may produce a variety of sampling representations based on finite frame theory and we discuss some applications.
For the entire collection see [Zbl 1485.65002].Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuityhttps://zbmath.org/1496.420322022-11-17T18:59:28.764376Z"Fedunyk-Yaremchuk, O. V."https://zbmath.org/authors/?q=ai:fedunyk-yaremchuk.oksana-volodymyrivna"Hembars'ka, S. B."https://zbmath.org/authors/?q=ai:hembarska.svitlana-borysivnaSummary: In this paper, we continue the study of approximation characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes \(B^{\Omega}_{p,\theta}\) in the space \(L_q\), \(1\leq p<q<\infty\), and also establish the exact-order estimates of approximation for these classes of functions in the space \(L_q\) by using linear operators satisfying certain conditions.Operators of harmonic analysis in grand variable exponent Morrey spaceshttps://zbmath.org/1496.420332022-11-17T18:59:28.764376Z"Kokilashvili, V."https://zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Meskhi, A."https://zbmath.org/authors/?q=ai:meskhi.alexanderSummary: The boundedness statements for the operators of Harmonic Analysis in grand variable exponent Morrey spaces are presented. Operators under consideration involve fractional and Calderón-Zygmund singular integral operators, Hardy-Littlewood maximal functions and commutators of singular integrals.Weak and strong estimates for rough Hausdorff type operator defined on \(p\)-adic linear spacehttps://zbmath.org/1496.420342022-11-17T18:59:28.764376Z"Volosivets, S. S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: For rough Hausdorff type operator defined on \(p\)-adic linear space \(Q^n_p\) and its commutator with symbol from Lipschitz space, we give sufficient conditions of their boundedness from one Lorentz space into another.Simon's OPUC Hausdorff dimension conjecturehttps://zbmath.org/1496.420352022-11-17T18:59:28.764376Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Guo, Shuzheng"https://zbmath.org/authors/?q=ai:guo.shuzheng"Ong, Darren C."https://zbmath.org/authors/?q=ai:ong.darren-cSummary: We show that the Szegő matrices, associated with Verblunsky coefficients \(\{{\alpha}_n\}_{n\in{{\mathbb{Z}}}_+}\) obeying \(\sum_{n = 0}^\infty n^{\gamma} |{\alpha}_n|^2 < \infty\) for some \({\gamma} \in (0,1)\), are bounded for values \(z \in \partial{\mathbb{D}}\) outside a set of Hausdorff dimension no more than \(1 - {\gamma}\). In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than \(1-{\gamma}\). This proves the OPUC Hausdorff dimension conjecture of \textit{B. Simon} [Orthogonal polynomials on the unit circle. Part 1: Classical theory. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.42020)].Orthogonal Dirichlet polynomialshttps://zbmath.org/1496.420362022-11-17T18:59:28.764376Z"Lubinsky, Doron S."https://zbmath.org/authors/?q=ai:lubinsky.doron-sSummary: Let \(\{ \lambda_j \}_{j=1}^{\infty}\) be a sequence of distinct positive numbers. Let w be a non-negative function, integrable on the real line. One can form orthogonal Dirichlet polynomials \(\{ \phi_n \}\) from linear combinations of \(\left\{ \lambda_j^{-it} \right\}_{j=1}^n\), satisfying the orthogonality relation
\[
\int_{-\infty}^{\infty}\phi_n(t) \overline{\phi_m(t)}w(t) dt=\delta_{mn}.
\]
Weights that have been considered include the arctan density \(w(t) =\frac{1}{\pi (1+t^2)}\); rational function choices of \(w\); \(w(t) =e^{-t}\); and \(w(t)\) constant on an interval symmetric about 0. We survey these results and discuss possible future directions.
For the entire collection see [Zbl 1485.65002].Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximationshttps://zbmath.org/1496.420372022-11-17T18:59:28.764376Z"Zaky, Mahmoud A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a"Abo-Gabal, Howayda"https://zbmath.org/authors/?q=ai:abo-gabal.howayda"Hafez, Ramy M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Doha, Eid H."https://zbmath.org/authors/?q=ai:doha.eid-hThe paper under review presents the main properties of a finite class of orthogonal polynomials with respect to the inverse gamma distribution over the positive real line called Romanovski-Bessel polynomials. More precisely, it introduces the related Romanovski-Bessel-Gauss-type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in weighted Sobolev space. It also addresses the relationship between such kinds of finite orthogonal polynomials and other classes of finite and infinite orthogonal polynomials.
Reviewer: M. Abdessadek Saib (Tebessa)Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh-Fourier serieshttps://zbmath.org/1496.420382022-11-17T18:59:28.764376Z"Gát, György"https://zbmath.org/authors/?q=ai:gat.gyorgy"Goginava, Ushangi"https://zbmath.org/authors/?q=ai:goginava.ushangiThe authors of this paper obtain results about the almost everywhere convergence and divergence of subsequences of Cesáro means with zero tending parameters of the Walsh-Fourier series. For this aim, let us consider the interval \(\mathbb{I}=[0,1)\) and let \(x=\sum_{n=0}^\infty\frac{x_n}{2^{n+1}}\) be the dyadic expansion of \(x\in\mathbb{I}\). Given \(n\in\mathbb{N}\), they consider its binary expansion \(n=\sum_{k=0}^\infty \varepsilon_k(n)2^k\), (where \(\varepsilon_k(n)=0\) or \(\varepsilon_k(n)=1\) for \(k\in\mathbb{N}\)) and denote \(|n|=\operatorname{Max}_{j\in\mathbb{N}}\{\varepsilon_j(n)\not=0\}\).
The \(n\)-th Walsh Paley function is defined as
\[
w_n(x)=(-1)^{\sum_{j=0}^\infty \varepsilon_j(n)x_j}, \ \ \ x\in\mathbb{I}.
\]
In this context, the Walsh-Dirichlet kernel and the Fejer kernel of the Walsh-Fourier series are given respectively by
\[
D_n(x)=\sum_{k=0}^{n-1}w_k(x)\ \ \ \text{and} \ \ \ K_n(x)=\frac{1}{n}\sum_{j=0}^{n-1}D_j(x).
\]
The partial sums of Walsh Fourier series are defined as \(S_m(f,x)=\sum_{j=0}^{m-1}\hat{f}(j)w_j(x)\), where \(\hat{f}(j)=\int_{\mathbb{I}}fw_j\) is the Walsh Fourier coefficient of the function \(f\in L^1(\mathbb{I})\).
This way, the \((C,\alpha_n)\) means of the Walsh Fourier series of the function \(f\) is given by
\[
\sigma_n^{\alpha_n}(f,x)=\frac{1}{A^{\alpha_n}_{n-1}}\sum_{j=1}^nA^{\alpha_n-1}_{n-j}S_j(f,x),
\]
where \(A^{\alpha_n}_{n-1}=\frac{(1+\alpha_n)\cdots(n+\alpha_n)}{n!}\), \(n\in\mathbb{N}\), \(\alpha_n\not=-1,-2,\dots\)
For a sequence \(\alpha=\{\alpha_n\}_{n\in\mathbb{N}}\) and \(K>0\), let us consider
\[
P_K(\alpha)=\left\{n\in\mathbb{N}:\frac{P(n,\alpha)}{n^{\alpha_n}}\leq K<\infty\right\}, \ \ \ \text{where} \ \ \ P(n,\alpha)=\sum_{i=0}^\infty\varepsilon_i(n)2^{i\alpha_n}.
\]
The authors introduce the weigthed version of the variation of an \(n\in\mathbb{N}\) by \(V(n,\alpha)=\sum_{i=0}^\infty|\varepsilon_i(n)-\varepsilon_{i+1}(n)|2^{i\alpha_n}\), with binary coefficients \(\{\varepsilon_k(n)\}_{k\in\mathbb{N}}\) and denote \(V_K(\alpha)=\left\{n\in\mathbb{N}:\frac{V(n,\alpha)}{n^{\alpha_n}}\leq K<\infty\right\}\).
So, they obtain the following results, among others:
\par i) Suppose \(\alpha_n\in(0,1)\). Let \(f\in L_1(\mathbb{I})\). Then, we have that \(\sigma_n^{\alpha_n}(f)\to f\), a.e., provided that \(V_k(\alpha)\ni n\to\infty\).
\par ii) Let \(f\in L_1(\mathbb{I})\) and \(\lim_{n\to\infty}\frac{V(n,\alpha)}{n^{\alpha_n}}=\infty\). Then, \(\lim_{n\to\infty}\frac{n^{\alpha_n}\sigma_n^{\alpha_n}(f,x)}{V(n,\alpha)}=0,\) a.e.
\par iii) Let \(f\in L_1(\mathbb{I})\). Then there exists a sequence \(\mu_j(f)\), such that for each subsequence of natural numbers with \(n_j\geq\mu_j(f)\), we have \(\sigma_{n_j}^{\alpha_{n_j}}(f)\to f\), a.e.
\par iv) For each sequence of natural numbers \(v_j\nearrow\infty\), there exists a function \(f\in L_1(\mathbb{I})\) and an another sequence of natural numbers with \(N_j\geq v_j\) for which we have the everywhere divergence of \(S_{N_j}(f)\).
\par v) Theorem: Let \(p>0\), then there exists a positive constant \(C_p\), such that, \(\| \operatorname{Sup}_{N\in\mathbb{N}} |f^\ast |K_{2^N}^{\alpha_N}| | \|_p \leq C_p \|f\|_{H_p}\), \(f\in H_p\), (\(H_p\) the Hardy space).
\par vi) Theorem: Let \(p>0\), then there exists a positive constant \(C_p\), such that, \(\| \operatorname{Sup}_{n\in P_K(\alpha)} |f^\ast |K_{n}^{\alpha_n}| | \|_p \leq C_p\| |f| \|_{H_p}\), \(f\in H_p\).
Reviewer: Iris Athamaica Lopez Palacios (Caracas)On the divergence sets of Fourier series in systems of characters of compact abelian groupshttps://zbmath.org/1496.420392022-11-17T18:59:28.764376Z"Oniani, G. G."https://zbmath.org/authors/?q=ai:oniani.georgi-g|oniani.giorgi-giglaSummary: For a class of character systems of compact abelian groups and for homogeneous Banach spaces \(B\) satisfying some additional regularity conditions, we prove the following alternative: either the Fourier series of an arbitrary function in \(B\) converges almost everywhere, or there exists a function in \(B\) whose Fourier series diverges everywhere. We also prove that the classes of divergence sets of Fourier series in such function systems in the above-mentioned spaces are closed under at most countable unions and contain all sets of measure zero. As corollaries, we obtain some well-known and new results on everywhere divergent Fourier series in the trigonometric system as well as in the Walsh and Vilenkin systems and their rearrangements.Exponential type bases on a finite union of certain disjoint intervals of equal lengthhttps://zbmath.org/1496.420402022-11-17T18:59:28.764376Z"Ghosh, Riya"https://zbmath.org/authors/?q=ai:ghosh.riya"Selvan, A. Antony"https://zbmath.org/authors/?q=ai:selvan.a-antonySummary: Let \(\Lambda\) be a sequence of distinct real numbers and \(\mathsf{T}=\{\epsilon_0,\dots ,\epsilon_{\nu -1}: \epsilon_i<\epsilon_{i+1}\}\) be a finite set of nonnegative integers. To each \((\Lambda ,\mathsf{T})\), we associate a system of exponentials
\[
\begin{aligned} \mathcal{E}(\Lambda,\mathsf{T}):=\left\{ x^{\epsilon_0}\mathrm{e}^{2\pi i\lambda x},x^{\epsilon_1}\mathrm{e}^{2\pi i\lambda x},\dots ,x^{\epsilon_{\nu -1}}\mathrm{e}^{2\pi i\lambda x}:\lambda \in \Lambda \right\}. \end{aligned}
\]
For a disconnected set \(\Omega\), the construction of a Riesz basis of exponential system \(\mathcal{E}(\Lambda,\mathsf{T})\) on \(L^2(\Omega)\) is generally a difficult problem. In the literature, the exponential Riesz basis problem is solved for certain disconnected sets \(\Omega\) for \(\nu =1\) and \(\epsilon_0=0\). It is well-known that this problem is equivalent to the construction of a complete interpolation set in the Paley-Wiener space \(\mathcal{PW}(\Omega)\). In this paper, we construct a complete interpolation set involving derivative samples in the Paley-Wiener space of a finite union of certain disjoint intervals of equal length. Furthermore, we provide an explicit sampling formula in this case.Toric symplectic geometry and full spark frameshttps://zbmath.org/1496.420412022-11-17T18:59:28.764376Z"Needham, Tom"https://zbmath.org/authors/?q=ai:needham.tom"Shonkwiler, Clayton"https://zbmath.org/authors/?q=ai:shonkwiler.claytonSummary: The collection of \(d \times N\) complex matrices with prescribed column norms and singular values forms an algebraic variety, which we refer to as a frame space. Elements of frame spaces -- i.e., frames -- are used to give robust signal representations, so that geometrical properties of frame spaces are of interest to the signal processing community. This paper is concerned with the question: what is the probability that a frame drawn at random from a given frame space has the property that any subset of \(d\) of its columns gives a basis for \(\mathbb{C}^d\)? We show that the probability is one, generalizing recent work of \textit{J. Cahill} et al. [SIAM J. Appl. Algebra Geom. 1, No. 1, 38--72 (2017; Zbl 1370.42023)]. To prove this, we show that frame spaces are related to highly structured objects called toric symplectic manifolds. As another application, we characterize the norm and spectral data for which the corresponding frame space has singularities, answering an open question in the literature.Frequency domain of weakly translation invariant frame MRAshttps://zbmath.org/1496.420422022-11-17T18:59:28.764376Z"Zhang, Zhihua"https://zbmath.org/authors/?q=ai:zhang.zhihua|zhang.zhihua.1This paper gives a characterization of the frequency domain of weakly translation invariant frame scaling functions with frequency domain \(\operatorname{supp} \widehat{\varphi}=G\). Based on this, it further characterizes convex and ball-shaped frequency domains of a bandlimited scaling function. When the frequency domain is convex and completely symmetric about the origin, then \(0 \in \operatorname{supp} \widehat{\varphi} \subset\left[-\frac{4}{3} \pi, \frac{4}{3} \pi\right]^d\) (this result cannot be improved). For the ball-shaped frequency domain in \(\mathbb{R}^d\) \((d>1)\), whether its center is the origin or not, its radius must satisfy:
\[
r \leq \max \left\{\frac{4}{3} \pi,\left(\sqrt{2+\frac{1}{4 d}}-\sqrt{\frac{1}{4 d}}\right) \pi\right\}.
\]
More importantly, these frequency domain characters are uniquely owned by frame scaling functions and not by orthogonal scaling functions: there does not exist an orthogonal scaling function with a ball-shaped frequency domain. If the frequency domain of the orthogonal scaling function is convex and completely symmetric about the origin, it must contain \([-\pi, \pi]^d\).
Reviewer: Pierluigi Vellucci (Roma)Order structures of \((\mathcal{D,E})\)-quasi-bases and constructing operators for generalized Riesz systemshttps://zbmath.org/1496.420432022-11-17T18:59:28.764376Z"Inoue, Hiroshi"https://zbmath.org/authors/?q=ai:inoue.hiroshiSummary: The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and \((\mathcal{D,E})\)-quasi bases for two fixed biorthogonal sequences \(\{\varphi_n\}\) and \(\{\Psi_n\}\). In a previous paper, we have studied the order structure of the set \(C_\varphi\) of all constructing operators for a generalized Riesz system \(\{\varphi_n\}\), and furthermore we have shown that the notion of generalized Riesz systems has a close relation with that of \((\mathcal{D,E})\)-quasi bases. For this reason, in this paper we define an order structure in the set \(\mathfrak{M}_{\varphi,\psi}\) of all pairs of dense subspaces \(\mathcal{D}\) and \(\mathcal{E}\) in \(\mathcal{H}\) such that \(\{\varphi_n\}\) and \(\{\psi_n\}\) are \((\mathcal{D,E})\)-quasi bases, and shall investigate the relationships between the order sets \(C_\varphi\), \(C_\psi\) and \(M_{\varphi,\psi}\). These results seem to be useful to find suitable constructing operators for each physical model.A note on the \(p\)-adic Kozyrev wavelets basishttps://zbmath.org/1496.420442022-11-17T18:59:28.764376Z"Arroyo-Ortiz, Edilberto"https://zbmath.org/authors/?q=ai:arroyo-ortiz.edilbertoThe theory of \(p\)-adic numbers has received considerable attention in several areas of mathematics, including number theory, algebraic geometry, algebraic topology and analysis, and several others. In this article, the authors have formulated a basis of \(p\)-adic wavelets for Sobolev-type spaces consisting of eigenvectors of certain pseudo-differential operators. The obtained results serve as an extension of the fundamental result due to \textit{S. Albeverio} and \textit{S. V. Kozyrev} [\(p\)-Adic Numbers Ultrametric Anal. Appl. 1, No. 3, 181--189 (2009; Zbl 1187.42030)] concerning the multidimensional basis of \(p\)-adic wavelets.
Reviewer: Azhar Y. Tantary (Srinagar)Free metaplectic wavelet transform in \(L^2(\mathbb{R}^n)\)https://zbmath.org/1496.420452022-11-17T18:59:28.764376Z"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmad"Qadri, Huzaifa L."https://zbmath.org/authors/?q=ai:qadri.huzaifa-l"Lone, Waseem Z."https://zbmath.org/authors/?q=ai:lone.waseem-zThis paper introduces the notion of free-metaplectic wavelet transform in \(L^2(\mathbb{R}^n)\). The paper studies some fundamental properties such as the orthogonality relation, inversion formula, characterization of range, and the homogeneous approximation property of the proposed transform. Heisenberg and Pitt's uncertainty inequalities associated with the free-metaplectic wavelet transform are established. The analysis of the double-window wavelet transform in the free metaplectic domains is done. The uncertainty principles associated with the free metaplectic wavelet transform are given. It is shown that the transform provides an efficient time-frequency representation of non-transient multidimensional signals.
Reviewer: Yilun Shang (Newcastle)A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\)https://zbmath.org/1496.430062022-11-17T18:59:28.764376Z"Bui, Huan Q."https://zbmath.org/authors/?q=ai:bui.huan-q"Randles, Evan"https://zbmath.org/authors/?q=ai:randles.evanSummary: In this article, we consider a class of functions on \(\mathbb{R}^d\), called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\). As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function \(P\), we construct a Radon measure \(\sigma_P\) on \(S=\{\eta \in\mathbb{R}^d:P(\eta)=1\}\) which is invariant under the symmetry group of \(P\). With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on \(\mathbb{Z}^d\) and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste
[\textit{E.~Randles} and \textit{L.~Saloff-Coste}, Rev. Mat. Iberoam. 33, No.~3, 1045--1121 (2017; Zbl 1377.42012)].Predual of \(M^{p,\alpha}(\mathbb{R}^d)\) spaceshttps://zbmath.org/1496.460072022-11-17T18:59:28.764376Z"Kpata, Berenger Akon"https://zbmath.org/authors/?q=ai:kpata.berenger-akonSummary: The space \( M^{p, \alpha} (\mathbb{R}^d)\) introduced by \textit{I. Fofana} [Afr. Mat., Sér. III 5, 53--76 (1995; Zbl 0885.42005)] is a subspace of the Wiener amalgam space of measures. In this note, we give a characterization of a predual space of this one.Completion by perturbationshttps://zbmath.org/1496.460102022-11-17T18:59:28.764376Z"Olevskii, Victor"https://zbmath.org/authors/?q=ai:olevskii.victorSummary: Any non-complete orthonormal system in a Hilbert space can be transformed into a basis by small perturbations.The embedding property of the scaling limit of modulation spaceshttps://zbmath.org/1496.460222022-11-17T18:59:28.764376Z"Chen, Jie"https://zbmath.org/authors/?q=ai:chen.jie.1|chen.jie.3|chen.jie|chen.jie.7|chen.jie.8|chen.jie.2|chen.jie.10|chen.jie.5|chen.jie.4|chen.jie.9|chen.jie.6"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Wang, Baoxiang"https://zbmath.org/authors/?q=ai:wang.baoxiangSummary: Modulation spaces \(M_{p , q}^s\) were introduced by Feichtinger in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [\textit{M.~Sugimoto} and \textit{B.-X. Wang}, Appl. Comput. Harmon. Anal. 53, 54--94 (2021; Zbl 1468.35187)] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [\textit{Á.~Bényi} and \textit{T.~Oh}, Appl. Comput. Harmon. Anal. 48, No.~1, 496--507 (2020; Zbl 1440.42101)], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as \(L^p\) spaces, Fourier \(L^p\) spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.Stable Gabor phase retrieval for multivariate functionshttps://zbmath.org/1496.460232022-11-17T18:59:28.764376Z"Grohs, Philipp"https://zbmath.org/authors/?q=ai:grohs.philipp"Rathmair, Martin"https://zbmath.org/authors/?q=ai:rathmair.martinSummary: In recent work [\textit{P.~Grohs} and \textit{M.~Rathmair}, Commun. Pure Appl. Math. 72, No.~5, 981--1043 (2019; Zbl 1460.94022)]
the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a function \(f\) from its spectrogram \(|\mathcal{G}f|\), where
\[
\mathcal{G}f(x,y)=\int_{\mathbb{R}^d} f(t) e^{-\pi|t-x|^2} e^{-2\pi i t\cdot y} \, dt, \quad x,y\in \mathbb{R}^d,
\] have been completely classified in terms of the disconnectedness of the spectrogram. These findings, however, were crucially restricted to the one-dimensional case (\(d=1\)) and therefore not relevant for many practical applications.
In the present paper we not only generalize the aforementioned results to the multivariate case but also significantly improve on them. Our new results have comprehensive implications in various applications such as ptychography, a highly popular method in coherent diffraction imaging.Functional calculus on BMO-type spaces of Bourgain, Brezis and Mironescuhttps://zbmath.org/1496.460322022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wenSummary: A nonlinear superposition operator \(T_g\) related to a Borel measurable function \(g: \mathbb{C}\to\mathbb{C}\) is defined via \(T_g(f):=g\circ f\) for any complex-valued function \(f\) on \(\mathbb{R}^n\). This article is devoted to investigating the mapping properties of \(T_g\) on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [\textit{J.~Bourgain} et al., J. Eur. Math. Soc. (JEMS) 17, No.~9, 2083--2101 (2015; Zbl 1339.46028)],
as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion and the continuity properties of \(T_g\) on these spaces are obtained.Pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness and their applicationshttps://zbmath.org/1496.460332022-11-17T18:59:28.764376Z"Li, Zi Wei"https://zbmath.org/authors/?q=ai:li.ziwei"Yang, Da Chun"https://zbmath.org/authors/?q=ai:yang.dachun.1|yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen|yuan.wen.1The nowadays well-known homogeneous spaces \(\dot{A}^s_{p,q} (\mathbb R^n)\) with \(A \in \{B,F \}\), \(s\in \mathbb R\) and \(0<p,q \le \infty\) have been modified in several ways. The smoothness \(s\), characterized by \(\{ 2^{js} \}^\infty_{j=0}\), is generalized by suitable sequences \(\{\sigma_j \}^\infty_{j=0}\) of positive numbers. Furthermore, \(\mathbb R^n\) is replaced by metric spaces \((X, d, \mu)\) with the metric \(d\) and the Borel measure \(\mu\) on the set \(X\), where the smoothness is expressed by so-called Hajłasz gradients. More recently, there is some type of discretization, called hyperbolic filling. The paper deals with spaces based on these ingredients and their relations, especially to \(\dot{A}^\sigma_{p,q} (\mathbb R^n)\).
Reviewer: Hans Triebel (Jena)A note on embedding inequalities for weighted Sobolev and Besov spaceshttps://zbmath.org/1496.460342022-11-17T18:59:28.764376Z"Saito, Hiroki"https://zbmath.org/authors/?q=ai:saito.hirokiLet \(H^d\), \(0<d<n\), be the Hausdorff capacity in \(\mathbb R^n\). The limiting embeddings
\[
\int_{\mathbb R^n} |f| \, d H^{n-k} \le c \, \|f \, | \dot{W}^k_1 (\mathbb R^n) \|, \qquad f \in \mathscr D(\mathbb R^n),
\]
\(1\le k <n\), \(k\in \mathbb N\), for the related homogeneous Sobolev spaces and its generalization
\[
\int_{\mathbb R^n} |f| \, d H^{n-s} \le c \,\|f \, | \dot{B}^s_{1,1} (\mathbb R^n) \|, \qquad f\in \mathscr D(\mathbb R^n),
\]
\(0<s<n\), for the related homogeneous Besov spaces go back to \textit{D. R. Adams} [Lect. Notes Math. 1302, 115--124 (1988; Zbl 0658.31009)] and \textit{J. Xiao} [Adv. Math. 207, No. 2, 828--846 (2006; Zbl 1104.46022)]. The paper deals with weighted generalizations of these assertions both for weighted Hausdorff capacities and weighted Sobolev-Besov spaces where the weights belong to the Muckenhoupt class~\(A_1\).
Reviewer: Hans Triebel (Jena)Sobolev's inequality in central Herz-Morrey-Musielak-Orlicz spaces over metric measure spaceshttps://zbmath.org/1496.460352022-11-17T18:59:28.764376Z"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuSummary: We give the boundedness of the Hardy-Littlewood maximal operator \(M_\lambda\), \(\lambda \geq 1\), on central Herz-Morrey-Musielak-Orlicz spaces \(\mathcal{H}^{\Phi,q,\omega}(X)\) over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials \(I_{\alpha,\tau}\), \(f\), \(\tau \geq 1\), \(\alpha > 0\) of functions in such spaces. As an application and example, we obtain the boundedness of \(M_\lambda\) and \(I_{\alpha,\tau}\) for double phase functionals \(\Phi\) such that \(\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}\), \(x \in X\), \(t \geq 0\). These results are new even for the doubling metric measure setting.Hardy classes and symbols of Toeplitz operatorshttps://zbmath.org/1496.470562022-11-17T18:59:28.764376Z"López-García, Marco"https://zbmath.org/authors/?q=ai:lopez-garcia.marco"Pérez-Esteva, Salvador"https://zbmath.org/authors/?q=ai:perez-esteva.salvadorSummary: The purpose of this paper is to study functions in the unit disk \(\mathbb D\) through the family of Toeplitz operators \(\{T_{\phi d\sigma_{t}}\}_{t\in[0,1)}\), where \(T_{\phi d\sigma_{t}}\) is the Toeplitz operator acting the Bergman space of \(\mathbb D\) and where \(d\sigma_t\) is the Lebesgue measure in the circle \(tS^1\). In particular for \(1\leq p < \infty\) we characterize the harmonic functions \(\phi\) in the Hardy space \(h^{p}(\mathbb D)\) by the growth in \(t\) of the \(p\)-Schatten norms of \(T_{\phi d\sigma_{t}}\). We also study the dependence in \(t\) of the norm operator of \(T_{ad\sigma_{t}}\) when \(a\in H^p_{at}\), the atomic Hardy space in the unit circle with \(1/2 < p \leq 1\).Conditions of invertibility for functional operators with shift in weighted Hölder spaceshttps://zbmath.org/1496.470662022-11-17T18:59:28.764376Z"Tarasenko, G."https://zbmath.org/authors/?q=ai:tarasenko.george-s"Karelin, O."https://zbmath.org/authors/?q=ai:karelin.oleksandrSummary: We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources.Intrinsic properties of strongly continuous fractional semigroups in normed vector spaceshttps://zbmath.org/1496.470682022-11-17T18:59:28.764376Z"Jones, Tiffany Frugé"https://zbmath.org/authors/?q=ai:fruge-jones.tiffany"Padgett, Joshua Lee"https://zbmath.org/authors/?q=ai:padgett.joshua-lee"Sheng, Qin"https://zbmath.org/authors/?q=ai:sheng.qinSummary: Norm estimates for strongly continuous semigroups have been successfully studied in numerous settings, but at the moment there are no corresponding studies in the case of solution operators of singular integral equations. Such equations have recently garnered a large amount of interest due to their potential to model numerous physically relevant phenomena with increased accuracy by incorporating so-called non-local effects. In this article, we provide the first step in the direction of providing such estimates for a particular class of operators which serve as solutions to certain integral equations. The provided results hold in arbitrary normed vector spaces and include the classical results for strongly continuous semigroups as a special case.
For the entire collection see [Zbl 1479.47003].Erdélyi-Kober fractional integral operators on ball Banach function spaceshttps://zbmath.org/1496.470742022-11-17T18:59:28.764376Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punThe author studies ball Banach function spaces and the Erdélyi-Kober fractional integral operators. The boundedness of the above operators on ball Banach function spaces is derived and proved. Furthermore, the boundedness of Erdélyi-Kober fractional integral operators on amalgan and Morrey spaces is also derived and proved.
Reviewer: James Adedayo Oguntuase (Abeokuta)Characterizations of pseudo-differential operators on \(\mathbb{S}^1\) based on separation-preserving operatorshttps://zbmath.org/1496.470752022-11-17T18:59:28.764376Z"Faghih, Zahra"https://zbmath.org/authors/?q=ai:faghih.zahra"Ghaemi, M. B."https://zbmath.org/authors/?q=ai:ghaemi.mohammad-bagherSummary: In this paper, we prove that a bounded pseudo-differential operator \(T_{\sigma}:L^p(\mathbb{S}^1)\rightarrow L^p(\mathbb{S}^1)\) for \(1\leq p<\infty\), is a separation-preserving operator and give a formula for its symbols \(\sigma\). Using these formulas, we give a new representation for the symbol of adjoint and products of two pseudo-differential operators.Riesz probability distributionshttps://zbmath.org/1496.600012022-11-17T18:59:28.764376Z"Hassairi, Abdelhamid"https://zbmath.org/authors/?q=ai:hassairi.abdelhamidFrom the cover of the book:\\
``Unique in the literature, this book provides an introductory, comprehensive and essentially self-contained exposition of the Riesz probability distribution on a symmetric cone and of its derivatives, with an emphasis on the case of the cone of positive definite symmetric matrices. \\
This distribution is an important generalization of the Wishart whose definition relies on the notion of generalized power. \\
Researchers in probability theory and harmonic analysis will find this book to be an important resource. \\
Given the connection between the Riesz probability distribution and the multivariate Gaussian samples with missing data, the book is also accessible and useful for statisticians.'' \\
\\
The preface finishes with:\\
``I hope that the book will be useful as a source of statements and applications of results in multivariate probability distributions and multivariate statistical analysis, as well as a reference to some material of harmonic analysis on symmetric cones adapted to the needs of researchers in these fields.'' \\
\\
The book is very large structured in Contents, Preface, Acknowledgment, 11 Chapters (with 45 subchapters), Bibliography (with 117 references), Index (with more than 70 items), Index of notations: \\
Chapter 1. Jordan algebras and symmetric cones -- Chapter 2. Generalized power -- Chapter 3. Riesz probability distributions -- Chapter 4. Riesz natural exponential families -- Chapter 5. Tweedie scale -- Chapter 6. Moments and constancy of regression -- Chapter 7. Beta Riesz probability distributions -- Chapter 8. Beta-Wishart distributions -- Chapter 9. Beta-hypergeometric distributions -- Chapter 10. Riesz-Dirichlet distributions -- Chapter 11. Riesz inverse Gaussian distribution \\
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The book can be very recommended all readers who are interested in this field.
Reviewer: Ludwig Paditz (Dresden)Correction to: ``Formulas of absolute moments''https://zbmath.org/1496.600162022-11-17T18:59:28.764376Z"Lin, Gwo Dong"https://zbmath.org/authors/?q=ai:lin.gwo-dong"Hu, Chin-Yuan"https://zbmath.org/authors/?q=ai:hu.chinyuanCorrection to the authors' paper [ibid. 83, No. 1, 476--495 (2021; Zbl 1459.60042)].Inclusion method of optimal constant with quadratic convergence for \(H_0^1\)-projection error estimates and its applicationshttps://zbmath.org/1496.652212022-11-17T18:59:28.764376Z"Kinoshita, Takehiko"https://zbmath.org/authors/?q=ai:kinoshita.takehiko"Watanabe, Yoshitaka"https://zbmath.org/authors/?q=ai:watanabe.yoshitaka"Yamamoto, Nobito"https://zbmath.org/authors/?q=ai:yamamoto.nobito"Nakao, Mitsuhiro T."https://zbmath.org/authors/?q=ai:nakao.mitsuhiro-tSummary: We present an interval inclusion method for optimal constants of second-order error estimates of \(H_0^1\)-projections to finite-degree polynomial spaces. These constants can be applied to error estimates of the Lagrange-type finite element method. Moreover, the proposed a priori error estimates are applicable to residual iteration techniques for the verification of solutions to nonlinear elliptic equations. Some numerical examples by the finite element method will be shown for comparison with other approaches, which confirm us the actual usefulness of the results in this paper for the numerical verification method for PDEs.Anisotropic elasticity and harmonic functions in Cartesian geometryhttps://zbmath.org/1496.740372022-11-17T18:59:28.764376Z"Labropoulou, D."https://zbmath.org/authors/?q=ai:labropoulou.d"Vafeas, P."https://zbmath.org/authors/?q=ai:vafeas.panayiotis"Dassios, G."https://zbmath.org/authors/?q=ai:dassios.georgeSummary: Linear elasticity in an isotropic space is a well-developed area of continuum mechanics. However, the situation is exactly opposite if the fundamental space exhibits anisotropic behavior. In fact, the area of linear anisotropic elasticity is not well developed at the quantitative level, where actual closed-form solutions are needed to be calculated. The present work aims to provide a little progress in this interesting branch of continuum mechanics. We provide a short review of isotropic elasticity in order to demonstrate in the sequel how the anisotropy modifies the final equations, via Hooke's and Newton's laws. The eight standard anisotropic structures are also reviewed for completeness. A simple technique is introduced that generates homogeneous polynomial solutions of the anisotropic equations in Cartesian form. In order to demonstrate how this technique is applied, we work out the case of cubic anisotropy, which is the simplest anisotropic structure, having three independent elasticities. This choice is dictated by the restricted number of calculations it requires, but it carries all the basic steps of the method. Isotropic elasticity accepts the differential representation of Papkovich, which expresses the displacement field in terms of a vector and a scalar harmonic function. Unfortunately, though, no such representation is known for the anisotropic elasticity, which can represent the anisotropic displacement field in terms of solutions of the anisotropic Laplacian, as also discussed in this work.
For the entire collection see [Zbl 1483.00042].Numerical analysis of resonances by a slab of subwavelength slits by Fourier-matching methodhttps://zbmath.org/1496.780202022-11-17T18:59:28.764376Z"Zhou, Jiaxin"https://zbmath.org/authors/?q=ai:zhou.jiaxin"Lu, Wangtao"https://zbmath.org/authors/?q=ai:lu.wangtaoThe novel method of the estimation of the Fourier transform based on noisy measurementshttps://zbmath.org/1496.940122022-11-17T18:59:28.764376Z"Galkowski, Tomasz"https://zbmath.org/authors/?q=ai:galkowski.tomasz"Pawlak, Miroslaw"https://zbmath.org/authors/?q=ai:pawlak.miroslawThe authors deal with the subject of analyzing the spectrum of signals associated with noise. They propose a method for estimating the frequency content of a signal derived from a nonparametric technique for function estimation. The mechanism used is based on orthogonal series expansions. Thus, the paper proposes a new integral version of nonparametric spectrum estimation using trigonometric series. Examples and numerical experiments are presented as well.
For the entire collection see [Zbl 1364.68015].
Reviewer: Liviu Goraş (Iaşi)Performance analysis for unconstrained analysis based approacheshttps://zbmath.org/1496.940142022-11-17T18:59:28.764376Z"Ge, Huanmin"https://zbmath.org/authors/?q=ai:ge.huanmin"Chen, Wengu"https://zbmath.org/authors/?q=ai:chen.wengu"Li, Dongfang"https://zbmath.org/authors/?q=ai:li.dongfang"Wu, Fengyan"https://zbmath.org/authors/?q=ai:wu.fengyanMultifractal characteristics of singular signalshttps://zbmath.org/1496.940172022-11-17T18:59:28.764376Z"Oświęcimka, Paweł"https://zbmath.org/authors/?q=ai:oswiecimka.pawel"Minati, Ludovico"https://zbmath.org/authors/?q=ai:minati.ludovicoStarting with locally Hölder continuous functions satisfying \[g(x+h)-g(x)\propto C\cdot h^{\alpha(x)},\] multifractal processes are characterized by a typically concave form of the ``singularity spectrum'' \(f(\alpha)\). The spectrum is defined as \(f(\alpha)=d(\{x\mid \alpha(x)\stackrel{!}{=}\alpha\})\), where \(d(\cdot)\) denotes the Hausdorff dimension of the respective set. Such processes are applied in such diverse areas as financial time series or medical applications [\textit{A. L. Karperien} et al., Banach Cent. Publ. 109, 23--45 (2016; Zbl 1355.28014)].
The present paper compares two spectrum estimation techniques, namely the ``Multifractal Detrended Fluctuation Analysis'' (MDFA) [\textit{J. W. Kantelhardt} et al., Physica A 316, No. 1--4, 87--114 (2002; Zbl 1001.62029)] and the ``Wavelet Leader'' (WL)-method [\textit{S. Jaffard}, Proc. Symp. Pure Math. 72, 91--151 (2004; Zbl 1093.28005)] on three data sets. Two of them exhibit typical recursive multiscale structures like Binomial cascades or similar Cantor-like constructions.
The third signal, however, exhibits clearly visible isolated singularities, and a comparision of MDFA/WL results with a standard wavelet analysis diagram -- clearly locating these isolated singularities -- indicates, that a multifractal interpretation of the signal, based on MDFA/WL might be misleading in this case.
For the entire collection see [Zbl 1491.93003].
Reviewer: Hans-Georg Stark (Aschaffenburg)Exact reconstruction of sparse non-harmonic signals from their Fourier coefficientshttps://zbmath.org/1496.940192022-11-17T18:59:28.764376Z"Petz, Markus"https://zbmath.org/authors/?q=ai:petz.markus"Plonka, Gerlind"https://zbmath.org/authors/?q=ai:plonka.gerlind"Derevianko, Nadiia"https://zbmath.org/authors/?q=ai:derevianko.nadiiaIn this paper, the authors propose a new method to reconstruct real non-harmonic Fourier sums, i.e. real signals which can be represented as sparse exponential sums of the form
\[
f(t)=\sum_{j=1}^{K}\gamma_{j}\cos(2\pi a_j t + b_j),
\]
from their Fourier coefficients. They assume that \(K\in\mathbb{N}\), \(\gamma_j\in (0,\infty)\), \((a_j, b_j) \in (0, \infty)\times[0, 2\pi )\), and that the frequency parameters \(a_j\) are pairwise distinct.
Their approach is based on two steps. The first one consists of reconstructing the non-\(P\)-periodic part of \(f\), employing a modification of the recently proposed AAA algorithm [\textit{Y. Nakatsukasa} et al., SIAM J. Sci. Comput. 40, No. 3, A1494--A1522 (2018; Zbl 1390.41015)]; the second step concerns the determination of possible \(P\)-periodic terms of \(f\). In particular, they prove that their method allows to uniquely determine \(f\) from at most \(2K+2\) of its Fourier coefficients.
Finally, the authors present two numerical experiments, which show that the considered reconstruction scheme provides very good reconstruction results even for small frequency gaps if \(P\) is chosen suitably. These results are compared with the ones obtained with a stabilized variant of Prony's method.
Reviewer: Mariarosaria Natale (Firenze)Sine series approximation of the mod function for bootstrapping of approximate HEhttps://zbmath.org/1496.940532022-11-17T18:59:28.764376Z"Jutla, Charanjit S."https://zbmath.org/authors/?q=ai:jutla.charanjit-s"Manohar, Nathan"https://zbmath.org/authors/?q=ai:manohar.nathanSummary: While it is well known that the sawtooth function has a point-wise convergent Fourier series, the rate of convergence is not the best possible for the application of approximating the mod function in small intervals around multiples of the modulus. We show a different sine series, such that the sine series of order \(n\) has error \(O(\epsilon^{2n+1})\) for approximating the mod function in \(\epsilon\)-sized intervals around multiples of the modulus. Moreover, the resulting polynomial, after Taylor series approximation of the sine function, has small coefficients, and the whole polynomial can be computed at a precision that is only slightly larger than \(-(2n+1)\log \epsilon\), the precision of approximation being sought. This polynomial can then be used to approximate the mod function to almost arbitrary precision, and hence allows practical CKKS-HE bootstrapping with arbitrary precision. We validate our approach by an implementation and obtain 100 bit precision bootstrapping as well as improvements over prior work even at lower precision.
For the entire collection see [Zbl 1493.94001].