Recent zbMATH articles in MSC 42https://zbmath.org/atom/cc/422024-02-15T19:53:11.284213ZWerkzeugOn multiplicative energy of subsets of varietieshttps://zbmath.org/1526.110052024-02-15T19:53:11.284213Z"Shkredov, Ilya D."https://zbmath.org/authors/?q=ai:shkredov.ilya-dSummary: We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.Positive bidiagonal factorization of tetradiagonal Hessenberg matriceshttps://zbmath.org/1526.150132024-02-15T19:53:11.284213Z"Branquinho, Amílcar"https://zbmath.org/authors/?q=ai:branquinho.amilcar"Foulquié-Moreno, Ana"https://zbmath.org/authors/?q=ai:foulquie-moreno.ana-pilar"Mañas, Manuel"https://zbmath.org/authors/?q=ai:manas.manuelThe authors study the tetradiagonal Hessenberg matrix of the form
\[
T = \begin{bmatrix} c_0 & 1 & 0 & \dots & \dots & \dots\\
b_1 & c_1 & 1& \ddots &&\\
a_2 & b_2 & c_2 & 1 & \ddots & &\\
0 & a_3 & b_3 & c_3 & 1 &\ddots&\\
\vdots & \ddots & \ddots & \ddots & \ddots & \ddots\\
\vdots && \ddots & \ddots & \ddots & \ddots \end{bmatrix},
\]
where \(a_n > 0\). They study the existence of a positive bidiagonal factorization (PBF) \(T = L_1 L_2 U\), where \(L_1\), \(L_2\), and \(U\) are bidiagonal matrices. They first summarize the theory pertaining to the tridiagonal case, emphasizing that a suitably shifted Jacobi matrix, denoted as \(\mathbf{J} + s \mathbf{I}\), exhibits oscillatory behavior. It is established that \(\mathbf{J}\) is oscillatory if and only if it possesses PBF properties. Then the authors use truncations and continued fractions to obtain their main results.
They prove the existence of a PBF for the tetradiagonal matrix in the finite-dimensional case. For the infinite-dimensional case, the authors prescribe a boundary condition in the presence of PBF, extrapolating the finite PBF to the semi-infinite case. This extrapolation shows the existence of a PBF when a specific nonnegative infinite continued fraction is unequivocally positive. Furthermore, they carry out an analysis of the oscillatory Toeplitz matrices, establishing their capability to admit PBF. Finally, they prove that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a PBF, but all the interior points of the ray do possess a PBF.
Reviewer: Tin Yau Tam (Reno)Fractional Fourier transform for space-time algebra-valued functionshttps://zbmath.org/1526.150252024-02-15T19:53:11.284213Z"Zayed, Mohra"https://zbmath.org/authors/?q=ai:zayed.mohra"El Haoui, Youssef"https://zbmath.org/authors/?q=ai:el-haoui.youssefSummary: The aim of this article is to introduce a fractional space-time Fourier transform (FrSFT) by generalizing the fractional Fourier transform for 16-dimensional space-time \(C \ell_{3,1} \)-valued signals over the domain of space-time (Minkowski space) \( \mathbb{R}^{3, 1} \). The primary analysis includes the investigation of fundamental properties such as the inversion, the Plancherel theorem, uniform continuity, and partial derivatives of the proposed transform. Using the space-time split, the Heisenberg uncertainty principle is established, and the FrSFT is employed to solve a partial differential equation in space-time analysis.On the sum of left and right circulant matriceshttps://zbmath.org/1526.150302024-02-15T19:53:11.284213Z"Lettington, Matthew C."https://zbmath.org/authors/?q=ai:lettington.matthew-c"Schmidt, Karl Michael"https://zbmath.org/authors/?q=ai:schmidt.karl-michaelLet \(A\) be a left circulant (also called Hankel) matrix and let \(B\) be a right circulant (also called Toeplitz) matrix. Then, the sum \(A+B\) is called a \textit{sum circulant matrix}. The paper is devoted to asymptotics of the sequences of powers of such matrices of odd orders. The main tool used is the discrete Fourier transform. The final result of this work is a characterization of eigenvalues of sum circulant matrices. Moreover, the authors present Moore-Penrose inverses of some particular sum circulant matrices.
Reviewer: Roksana Słowik (Gliwice)Spectral multipliers for functions of fixed \(K\)-type on \(L^p(\mathrm{SL}(2,\mathbb{R})\)https://zbmath.org/1526.220062024-02-15T19:53:11.284213Z"Ricci, Fulvio"https://zbmath.org/authors/?q=ai:ricci.fulvio.1"Wróbel, Błażej"https://zbmath.org/authors/?q=ai:wrobel.blazej-janSummary: We prove an \(L^p\) spectral multiplier theorem for functions of the \(K\)-invariant sublaplacian \(L\) acting on the space of functions of fixed \(K\)-type on the group \(SL(2,\mathbb{R})\). As an application we compute the joint \(L^p(SL(2,\mathbb{R}))\) spectrum of \(L\) and the derivative along \(K\).On the differentiation of random measures with respect to homothecy invariant convex baseshttps://zbmath.org/1526.280012024-02-15T19:53:11.284213Z"Chubinidze, Kakha"https://zbmath.org/authors/?q=ai:chubinidze.kakha-a"Oniani, Giorgi"https://zbmath.org/authors/?q=ai:oniani.giorgi-giglaSummary: For every homothecy invariant convex density differentiation basis \(B\) in \(\mathbb{R}^d\), we characterize sequences of weights \(w=(w_j)_{j\in \mathbb{N}}\) for which the random measures \(\mu_{w,\theta}=\sum_{j=1}^\infty w_j \delta_{\theta_j}\) are differentiable with respect to the basis \(B\) for almost every selection of a sequence of points \(\theta_1,\theta_2,\ldots\) from the unit cube \([0,1]^d\).F. Wiener's trick and an extremal problem for \(H^p\)https://zbmath.org/1526.300682024-02-15T19:53:11.284213Z"Brevig, Ole Fredrik"https://zbmath.org/authors/?q=ai:brevig.ole-fredrik"Grepstad, Sigrid"https://zbmath.org/authors/?q=ai:grepstad.sigrid"Instanes, Sarah May"https://zbmath.org/authors/?q=ai:instanes.sarah-maySummary: For \(0<p \le \infty\), let \(H^p\) denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the \(k\)th Taylor coefficient of a function \(f \in H^p\) which satisfies \(\Vert f\Vert_{H^p}\le 1\) and \(f(0)=t\) for some \(0 \le t \le 1\). In particular, we provide a complete solution to this problem for \(k=1\) and \(0<p<1\). We also study F. Wiener's trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.Well-posedness of differential equations on \(B^s_{p, q}(\mathbb{R}; X)\) and \(F^s_{p, q}(\mathbb{R}; X)\)https://zbmath.org/1526.340382024-02-15T19:53:11.284213Z"Bu, Shangquan"https://zbmath.org/authors/?q=ai:bu.shangquan"Zhong, Yuchen"https://zbmath.org/authors/?q=ai:zhong.yuchenIn the paper under review, the authors apply the Fourier multipliers theorems on the spaces \(B_{p,q}^{s}({\mathbb R};X),\) \(F_{p,q}^{s}({\mathbb R};X),\) \(B_{p,q}^{s,\omega}({\mathbb R};X),\) \(F_{p,q}^{s,\omega}({\mathbb R};X)\) and the Carleman transform in order to study the well-posedness of the abstract Cauchy problems of first order and the abstract Cauchy problems of second order. The obtained results are non-trivial and interesting.
Reviewer: Marko Kostić (Novi Sad)A reverse Ozawa-Rogers estimatehttps://zbmath.org/1526.350892024-02-15T19:53:11.284213Z"Huang, Yi C."https://zbmath.org/authors/?q=ai:huang.yichi|huang.yi-chieh|huang.yi-c|huang.yichao|huang.yichun|huang.yicheng|huang.yicong|huang.yichenSummary: We provide a reverse bilinear estimate for the one-dimensional Klein-Gordon equation that complements a result of Ozawa and Rogers. The proof relies on the cosine formula for frequency vectors adapted to the Klein-Gordon equation.A Green function characterization of uniformly rectifiable sets of any codimensionhttps://zbmath.org/1526.351282024-02-15T19:53:11.284213Z"Feneuil, Joseph"https://zbmath.org/authors/?q=ai:feneuil.joseph"Li, Linhan"https://zbmath.org/authors/?q=ai:li.linhanThe paper under review deals with the study of a characterization of the uniform rectifiability of sets. More precisely, the authors show a unified characterization of uniformly rectifiable sets of any codimension by means of a Carleson estimate of the second derivatives of the Green function. In particular, in case of domains with boundary of codimension 1, the present paper generalizes a result of [\textit{J. Azzam}, ``Harmonic measure and the analyst's traveling salesman theorem'', Preprint, \url{arXiv:1905.09057}] for the Laplacian, to general elliptic operators.
Reviewer: Paolo Musolino (Padova)Global solutions to the damped MHD systemhttps://zbmath.org/1526.352682024-02-15T19:53:11.284213Z"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaoping"Chen, Zhi-Min"https://zbmath.org/authors/?q=ai:chen.zhiminA system of inviscid conducting fluid coupled with magnetic field equations is considered in \(\mathbb{R}^n\) with linear damping terms (with equal damping coefficients) in both equations for the fluid velocity \(u\) and the magnetic field \(b\). If sufficiently regular \(u\) and \(b\) are initially either small or close to each other, the global-in-time existence result is proved in the Besov space \(B^s_{p,1}\) whenever \(n\ge 2\), \(1<p<\infty\), \(s\ge \frac{n}{p}+1\), with \(C\cap L^1\) dependence on time in \((0,\infty)\).
Reviewer: Piotr Biler (Wrocław)On a robust stability criterion in the subdiffusion equation with Caputo-Dzherbashian fractional derivativehttps://zbmath.org/1526.353002024-02-15T19:53:11.284213Z"Temoltzi-Ávila, R."https://zbmath.org/authors/?q=ai:temoltzi-avila.raulSummary: This paper presents a robust stability criterion for the subdiffusion equation with Caputo-Dzherbashian fractional derivative. The criterion is obtained by extending the concept of stability under constant-acting perturbations applied to systems of differential equations of integer order. It is assumed that the subdiffusion equation admits external sources that are represented by Fourier series. The robust stability criterion makes it possible to ensure that the solution of the subdiffusion equation, as well as its Caputo-Dzherbashian fractional derivative and its first partial derivative with respect to the longitudinal axis, are bounded.Fractional Calderón problems and Poincaré inequalities on unbounded domainshttps://zbmath.org/1526.353232024-02-15T19:53:11.284213Z"Railo, Jesse"https://zbmath.org/authors/?q=ai:railo.jesse"Zimmermann, Philipp"https://zbmath.org/authors/?q=ai:zimmermann.philippSummary: We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.Multilinear oscillatory integrals and estimates for coupled systems of dispersive PDEshttps://zbmath.org/1526.353532024-02-15T19:53:11.284213Z"Bergfeldt, Aksel"https://zbmath.org/authors/?q=ai:bergfeldt.aksel"Rodríguez-López, Salvador"https://zbmath.org/authors/?q=ai:rodriguez-lopez.salvador"Rule, David"https://zbmath.org/authors/?q=ai:rule.david-j"Staubach, Wolfgang"https://zbmath.org/authors/?q=ai:staubach.wolfgangSummary: We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.Special Weber transform with nontrivial kernelhttps://zbmath.org/1526.353542024-02-15T19:53:11.284213Z"Gorshkov, A. V."https://zbmath.org/authors/?q=ai:gorshkov.anton-v|gorshkov.alexandr-vasilievich|gorshkov.alexey-vSummary: We study the Weber integral transforms \(W_{k,k\pm1}\), which have a nontrivial kernel, so that the spectral expansion contains not only the continuous part of the spectrum but also the zero eigenvalue corresponding to the kernel. The inversion formula, the spectral decomposition, and the Plancherel-Parseval equality are derived. These transforms are used in an explicit formula for the solution of the classical nonstationary Stokes problem on the flow past a circular cylinder.Self-affine 2-attractors and tileshttps://zbmath.org/1526.370212024-02-15T19:53:11.284213Z"Zaitseva, Tatyana I."https://zbmath.org/authors/?q=ai:zaitseva.tatyana-i"Protasov, Vladimir Yu."https://zbmath.org/authors/?q=ai:protasov.vladimir-yuSummary: We study two-digit attractors (2-attractors) in \(\mathbb{R}^d\), which are self-affine compact sets defined by two affine contractions with the same linear part. They have widely been studied in the literature under various names (integer self-affine 2-tiles, twindragons, two-digit tiles, 2-reptiles and so on) due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in discrete geometry and in number theory. We obtain a complete classification of isotropic 2-attractors in \(\mathbb{R}^d\) and show that all of them are pairwise homeomorphic but not diffeomorphic. In the general, nonisotropic, case we prove that a 2-attractor is uniquely defined by the spectrum of the dilation matrix up to affine similarity. We estimate the number of different 2-attractors in \(\mathbb{R}^d\) by analysing integer unitary expanding polynomials with free coefficient \(\pm2\). The total number of such polynomials is estimated using the Mahler measure. We present several infinite series of such polynomials. For some 2-attractors their Hölder exponents are found. Some of our results are extended to attractors with an arbitrary number of digits.On the Lebesgue constantshttps://zbmath.org/1526.410082024-02-15T19:53:11.284213Z"Kushpel, A. K."https://zbmath.org/authors/?q=ai:kushpel.alexander-kSummary: We present the solution of a classical problem of approximation theory about the sharp asymptotics of Lebesgue constants or the norms of Fourier-Laplace projections on the real sphere \({\mathbb{S}}^d ,\) in complex \(P^d (\mathbb{C} )\) and quaternionic \(P^d (\mathbb{H} )\) projective spaces, and in the Cayley elliptic plane \(P^{16}(Cay)\). In particular, these results supplement the sharp asymptotics established by Fejer (1910) in the case of \({\mathbb{S}}^1\) and by Gronwall (1914) in the case of \({\mathbb{S}}^2 .\)Geometric harmonic analysis IV. Boundary layer potentials in uniformly rectifiable domains, and applications to complex analysishttps://zbmath.org/1526.420012024-02-15T19:53:11.284213Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the fourth in a series of five volumes, at the confluence of Harmonic Analysis,
Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically
branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles:
Volume~I: A Sharp Divergence Theorem with Nontangential Pointwise Traces;
Volume~II: Function Spaces Measuring Size and Smoothness on Rough Sets;
Volume~III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering;
Volume~IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis;
Volume~V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
The main objective of the series is to produce tools that can treat efficiently boundary value problems
for elliptic systems in inclusive geometric settings, beyond the category of Lipschitz domains.
In this fourth volume, the bulk of the results amounts to a versatile Calderón-Zygmund
theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries.
The picture that emerges is that Calderón-Zygmund theory is a multi-faceted body of results
aimed at describing how singular integral operators behave in many geometric and analytic settings.
Applications to Complex Analysis in several variables are subsequently presented, starting from the realization that many natural integral operators in this setting, such as the Bochner-Martinelli operator, are actual particular cases of double-layer potential operators associated with the complex Laplacian. What follows is a concise description of the contents of each chapter.
Chapter 1 focuses on singular integral operators (SIOs) of boundary layer type on Lebesgue and Sobolev spaces.
Generic Calderón-Zygmund convolution-type SIOs [Volume~III, Chapter~2] are not expected to induce well-defined
mappings on Sobolev spaces on uniformly rectifiable (UR) sets, as this requires a special algebraic structure of
their integral kernels. Topics treated in this chapter include the history and physical interpretations of the
classical harmonic layer potentials, ``tangential'' singular integral operators, volume and integral operators of
boundary layer type associated with a given open set of locally finite perimeter and a given weakly elliptic system,
a multitude of relevant examples and alternative points of view, a rich function theory of Calderón-Zygmund type for
boundary layer potentials associated with a given weakly elliptic system and an open set with a uniformly rectifiable boundary,
the interpretation of the Cauchy and Cauchy-Clifford operators as double-layer potential operators,
and how to modify boundary layer potential operators to increase the class of functions to which they may be applied.
Chapter 2 concentrates on layer potential operators acting on Hardy, BMO, VMO, and Hölder spaces defined on boundaries of UR domains.
A fundamental aspect of this analysis is that a special algebraic structure is required of the integral kernel for a singular integral operator to map either of these spaces into itself and the brand of Divergence Theorem produced in Volume I plays a significant part.
In fact, the same type of philosophy prevails in relation to the action of double-layer potential operators on Calderón,
Morrey-Campanato, and Morrey spaces discussed in Chapter 3, and also for the action of double layer potential operators on Besov and
Triebel-Lizorkin spaces, treated in Chapter 4.
Chapter 5 addresses the following basic question: describe the most general classes of singular integral operators
on the boundary of an arbitrary given UR domain $\Omega\subset{\mathbb{R}}^n$ which map Hardy, BMO, VMO, Hölder, Besov, and
Triebel-Lizorkin spaces defined on $\partial\Omega$ boundedly into themselves. The authors provide an answer through the
introduction of what they call generalized double-layer operators. They also take a look at Riesz transforms
from the point of view of generalized double layers.
In Chapter 6 the authors develop a theory of boundary layer potentials associated with the Stokes system of linear hydrostatics,
and related topics. Among other things, they establish Green-type formulas, derive mapping properties for the aforementioned
boundary layer potential operators, and prove Fatou-type results, in settings that are sharp from a geometric/analytic
point of view. Once again, the brand of Divergence Theorem discussed in Volume~I plays a prominent role in carrying out this program.
Chapter 7 contains a multitude of applications of the body of results developed so far in the area of Geometric Harmonic Analysis
to the field of Complex Analysis of Several Variables. As is well known, Complex Analysis, Geometric
Measure Theory and Harmonic Analysis interface tightly in the complex plane. However, this rich interplay
between these branches of mathematics has been considerably less explored in the higher-dimensional setting,
involving several complex variables. The main goal of the current chapter is to further the present understanding
of this aspect. Themes covered include CR-functions and differential forms on boundaries of sets of locally finite perimeter,
integration by parts formulas involving the \(d\)-bar operator on sets of locally finite perimeter,
the Bochner-Martinelli integral operator, a sharp version of the Bochner-Martinelli-Koppelman formula,
the Extension Problem for CR-functions in a variety of spaces on boundaries of Ahlfors regular (and UR) domains.
Chapter 8 focuses on the study of Hardy spaces for certain second-order weakly elliptic operators in the complex plane, such as
the Bitsadze operator in the unit disk. The purpose of the chapter is to characterize the space of null solutions and to
identify precisely the corresponding spaces of boundary traces.
Reviewer: Mohammed El Aïdi (Bogotá)Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions in the space \(B_{ \infty ,1}\)https://zbmath.org/1526.420022024-02-15T19:53:11.284213Z"Hembars'ka, S. B."https://zbmath.org/authors/?q=ai:hembarska.svitlana-borysivna"Zaderei, P. V."https://zbmath.org/authors/?q=ai:zaderei.p-vSummary: Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and many variables in the space \(B_{ \infty ,1}.\)On the best simultaneous approximation of functions in the Bergman space \(B_2\)https://zbmath.org/1526.420032024-02-15T19:53:11.284213Z"Khuromonov, Kh. M."https://zbmath.org/authors/?q=ai:khuromonov.khuromon-mamadamonovichSummary: Extreme problems related to the best simultaneous polynomial approximation of analytic functions in the unit disc belonging to the Bergman space \({{B}_2}\) are studied. Here, a number of exact theorems and the exact values of the upper bounds of the best simultaneous approximations of functions and their consecutive derivatives by polynomials and their corresponding derivatives on some classes of complex functions belonging to the Bergman space \({{B}_2}\) are obtained.Approximation of the classes of periodic functions of one and many variables from the Nikol'skii-Besov and Sobolev spaceshttps://zbmath.org/1526.420042024-02-15T19:53:11.284213Z"Romanyuk, A. S."https://zbmath.org/authors/?q=ai:romanyuk.anatolii-sergiiovych"Yanchenko, S. Ya."https://zbmath.org/authors/?q=ai:yanchenko.serhii-yaSummary: We establish the exact-order estimates for the best orthogonal trigonometric approximations of the Nikol'skii-Besov classes \({B}_{1,\theta}^r ( \mathbb{T}^d)\), \(1 \leq \theta \leq \infty ,\) of periodic functions of one and many variables with predominant mixed derivative in the space \(B_{ \theta , 1} ( \mathbb{T}^d)\). In the multidimensional case, \( d \geq 2,\) we establish the exact-order estimates for the approximations of the indicated classes of functions by their step-hyperbolic Fourier sums and determine the orders of orthoprojection widths in the same space. The behaviors of the corresponding approximation characteristics of the Sobolev classes \({W}_{1,\alpha}^r ( \mathbb{T}^d )\) with \(d \in\{1, 2\}\) are also investigated.Almost everywhere divergence of Cesàro means with varying parameters of Walsh-Fourier serieshttps://zbmath.org/1526.420052024-02-15T19:53:11.284213Z"Goginava, Ushangi"https://zbmath.org/authors/?q=ai:goginava.ushangiSummary: In the presented paper, we are going to solve the problem related to the almost everywhere divergence of the \((C, \alpha_n)\) means of Walsh-Fourier series. Namely, we establish that for every sequence \((\alpha_n : n \in \mathbb{P})\) that tends to zero arbitrarily slowly, there exists an integrable function \(f\) for which \((C, \alpha_n)\) means of Walsh-Fourier series is divergent almost everywhere.A sufficient condition for uniform convergence of trigonometric series with \(p\)-bounded variation coefficientshttps://zbmath.org/1526.420062024-02-15T19:53:11.284213Z"Kubiak, Mateusz"https://zbmath.org/authors/?q=ai:kubiak.mateusz"Szal, Bogdan"https://zbmath.org/authors/?q=ai:szal.bogdanFor any real \(c>0\), the uniform convergence of the series
\[
\sum_{n=1}^{\infty}b_{n}\sin (cnx),\quad \sum_{n=1}^{\infty}a_{n}\cos (cnx),\quad \text{and}\quad \sum_{n=1}^{\infty}c_{n}e^{icnx}
\]
is studied.
The authors show that these series converge uniformly if (among other conditions) the sequence \((b_{n})\) belongs to the class \(GM (p, 3\beta(q), r)\), where \(q\geq 1\), \(p>1\) and \(r\in \mathbb{N}\). We mention here that the latest class has been defined earlier by the same authors [Bull. Belg. Math. Soc. - Simon Stevin 27, No. 1, 89--110 (2020; Zbl 1453.42004)]. It is vitally to say that the new results are proved under weakened assumptions in case \(p>1\).
Reviewer: Xhevat Z. Krasniqi (Prishtina)\(L^1\) convergence of Fourier transformshttps://zbmath.org/1526.420072024-02-15T19:53:11.284213Z"Liflyand, E."https://zbmath.org/authors/?q=ai:liflyand.elijahSummary: This is the first attempt to generalize the problem of \(L^1\) convergence of trigonometric series to the non-periodic case. We extend one of the most general results and then show the way how to derive its prototype from the obtained extension.
For the entire collection see [Zbl 1515.42003].Boas and Titchmarsh type theorems for generalized Lipschitz classes and \(q\)-Bessel Fourier transformhttps://zbmath.org/1526.420082024-02-15T19:53:11.284213Z"Volosivets, S. S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevich"Krotova, Yu. I."https://zbmath.org/authors/?q=ai:krotova.yu-iSummary: Necessary and sufficient conditions for a function \(f\) to belong to the generalized Lipschitz classes \(H^{m,\omega}_{q,\nu}\) and \(h^{m,\omega}_{q,\nu}\) for fractional \(m\) are given in terms of its \(q\)-Bessel-Fourier transform \(\mathcal F_{q,\nu}(f)\). Dual results are considered as well. An analog of the Titchmarsh theorem for fractional-order differences is proved.On (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth modulihttps://zbmath.org/1526.420092024-02-15T19:53:11.284213Z"Chen, Shaolin"https://zbmath.org/authors/?q=ai:chen.shaolin"Hamada, Hidetaka"https://zbmath.org/authors/?q=ai:hamada.hidetakaSummary: The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in [\textit{D. Kalaj}, Trans. Am. Math. Soc. 372, No. 6, 4031--4051 (2019; Zbl 1422.30002)]. Next, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established, which give analogies and extensions of a result in [\textit{G. H. Hardy} and \textit{J. E. Littlewood}, J. Reine Angew. Math. 167, 405--423 (1932; JFM 58.0333.03)]. Furthermore, we establish a Fejér-Riesz type inequality on pluriharmonic functions in the Euclidean unit ball in \(\mathbb{C}^n\), which extends the main result in [\textit{P. Melentijević} and \textit{V. Božin}, Potential Anal. 54, No. 4, 575--580 (2021; Zbl 1460.31005)]. Additionally, we also discuss the Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and extend the corresponding results in \textit{K. M. Dyakonov} [Acta Math. 178, No. 2, 143--167 (1997; Zbl 0898.30040)], \textit{G. H. Hardy} and \textit{J. E. Littlewood}, [Math. Z. 34, 403--439 (1931; JFM 57.0476.01)], \textit{K. M. Dyakonov} [Adv. Math. 187, No. 1, 146--172 (2004; Zbl 1056.30018)] and \textit{M. Pavlović} [Rev. Mat. Iberoam. 23, No. 3, 831--845 (2007; Zbl 1148.31003)].Existence of dense subsystems with lacunarity property in orthogonal systemshttps://zbmath.org/1526.420102024-02-15T19:53:11.284213Z"Limonova, Irina V."https://zbmath.org/authors/?q=ai:limonova.irina-vLet \(\Psi=\{\phi_k\}_{k=1}^N\) be an orthogonal system of functions on a probability space \((X,\mu)\). Put \(\langle N\rangle\equiv\{1,2,\dots,N\}\), and for \(\Lambda\subset\langle N\rangle\) let \(S_\Lambda\) be the operator defined by \(S_\Lambda(\{a_k\}_{k\in\Lambda})=\sum_{k\in\Lambda}a_k\phi_k(x)\). Put \(\Psi_\Lambda=\{\phi_k\}_{k\in\Lambda}\). Let \(\delta\in(0,1)\), and \(\{\xi_i(\omega)\}_{i=1}^N\) be a system of independent random variables on a probability space \((\Omega,\nu)\) such that \(\xi_i(\omega)=0\) or \(1\) and \(E\xi_i=\delta\) \((1\leq i\leq N)\). For \(\omega\in\Omega\), let \(\{\xi_i(\omega)\equiv\Lambda(\omega,N)\equiv\{i\in\langle N\rangle:\xi_i(\omega)=1\}\).
Below the author considers the scale of Orlicz spaces \(L_{\psi_\alpha}\), where \(\alpha>0\),
\[
\psi_\alpha(t)=t^2\frac{\ln^\alpha(e+|t|)}{\ln^\alpha(e+1/|t|)},\text{ and } \|f\|_{\psi_\alpha}=\inf\{\lambda>0:\int_X\psi_\alpha(\frac{f(x)}{\lambda})d\mu\leq1\}.
\]
\textit{J. Bourgain} [Acta Math. 162, No. 3--4, 227--245 (1989; Zbl 0674.43004)] solved the problem of the existence of \(p\)-lacunary (\(p>2\)) subsystems of size \(N^{2/p}\) in an arbitrary uniformly bounded orthogonal system \(\{\phi_k\}_{k=1}^N\). Using a modification of the method in [Bourgain, loc. cit.], \textit{B. S. Kashin} and \textit{I. V. Limonova} [Proc. Steklov Inst. Math. 311, 152--170 (2020; Zbl 1457.42015); translation from Tr. Mat. Inst. Steklova 311, 164--182 (2020)] obtained the following:
Theorem A.
Let \(\alpha>0\) and \(\rho>0\) be fixed, and let \(\Psi=\{\phi_k\}_{k=1}^N\) be an arbitrary orthogonal system with the property
\[
\|\phi_k\|\leq1,\ k=1,2,\dots,N.
\]
Then for a random set \(\Lambda=\Lambda(\omega)\) generated by a set of random variables \(\{\xi_i(\omega)\}_{i=1}^N\) such that \(\mathrm{E}\xi_i=\delta=(\log_2(N+3))^{-\rho}\), \(1\leq i\leq N\), the inequality
\[
\|S_\Lambda:\ell_\infty(\Lambda)\rightarrow L_{\psi_\alpha}(X)\|\leq K(\alpha,\rho)|\Lambda|^{1/2}((\log_2(N+3))^{\alpha/2-\rho/4}+1)
\]
holds with probability greater than \(1-C(\rho)N^{-\alpha}\).
In this paper, the author obtains a generalization of Theorem A to \(S_\Lambda\) acting from the space \(\ell_2(\Lambda)\) to \(L_{\psi_\alpha}(X)\), and to functions \(\phi_k\) satisfying only the weaker condition \(\|\phi_k\|_p\leq1\), \(1\leq k\leq N\), where \(p>4\). Also, this paper is deeply related to [Kashin and Limonova, loc. cit.].
Reviewer: Enji Sato (Yamagata)On basicity of a certain trigonometric system in a weighted Lebesgue spacehttps://zbmath.org/1526.420112024-02-15T19:53:11.284213Z"Guliyeva, A. E."https://zbmath.org/authors/?q=ai:guliyeva.aysel-eSummary: In this paper one perturbed system of exponents \(1\cup\left\{e^{\pm i\lambda_nt}\right\}_{n\in\mathbb{N}}\) is considered, where \(\lambda_n=\sqrt{n^2+\alpha n+\beta}\), \(\forall\,n\in\mathbb{N} \). A weighted Lebesgue space \(L_{p,w}\left(-\pi,\pi\right)\), \(1<p<+\infty\) is considered, where \(w:\left[-\pi,\pi\right]\to\left[0,+\infty\right]\) is a weight function. A sufficient condition for the basicity of this system depending on the parameters \(\alpha;\beta\in\mathbb{R}\) in \(L_{p,w}\left(-\pi,\pi\right)\) is founded for the case when weight \(w\) satisfies the Muckenhoupt condition.Stepanov and Weyl classes of \(c\)-almost periodic type functionshttps://zbmath.org/1526.420122024-02-15T19:53:11.284213Z"Ounis, Hadjer"https://zbmath.org/authors/?q=ai:ounis.hadjer"Sepulcre, Juan Matías"https://zbmath.org/authors/?q=ai:sepulcre.juan-matiasAuthors' abstract: As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of \(c\)-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every \(c\)-almost periodic function in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) is also \(c^m\)-almost periodic in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) for each non-zero integer number \(m\). This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complex functions defined on vertical strips in the complex plane.
Reviewer: Sorin Gheorghe Gal (Oradea)Determining the jump of a function of \(m\)-harmonic bounded variation by its Fourier serieshttps://zbmath.org/1526.420132024-02-15T19:53:11.284213Z"Kelzon, A. A."https://zbmath.org/authors/?q=ai:kelzon.a-aSummary: In this article, the well-known formula for determining the jump of a periodic function using the derivative of the partial sums of its Fourier series is extended to a new class of functions.Convergence of Vilenkin-Fourier series in variable Hardy spaceshttps://zbmath.org/1526.420142024-02-15T19:53:11.284213Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferencLet \(p(.):[0,1)\to (0,\infty)\) be a variable exponent function and for any measurable set \(A\subset [0,1)\), let us denote
\[
p_{-}(A)=\operatorname*{essinf}_{x\in A}p(x)\quad \text{and} \quad p_{+}(A)=\operatorname*{esssup}_{x\in A}p(x).
\]
Particularly, if \(A=[0,1)\) then \(p\_(A)\) and \(p_{+}(A)\) are denoted as \(p\-\) and \(p_{+}\). Let us denote by \(\mathcal{P}\) the collection of all variable exponent \(p(.)\), such that, \(0<p\_ \leq p_{+}\). So, the author defines by \(C^{\log}\) the set of all functions \(p(.)\in \mathcal{P}\), satisfying the following property: it exists a positive constant \(C_{\log}(p)\), such that, for any \(x,y\in [0,1)\) we have,
\[
|p(x)-p(y)|\leq \frac{C_{\log}(p)}{\text{log}(e+1/|x-y|)}.
\]
The above inequality is called the log-Hölder continuous condition. Now, let \((p_n)_{n\in\mathbb{N}}\) be a sequence of natural numbers with entries at least 2. Suppose that \((p_n)_{n\in\mathbb{N}}\) is bounded. Then, let us introduce the notations \(P_0=1\) and \(P_{n+1}=\prod_{k=0}^{n}p_k\), \(k\in\mathbb{N}\), and let \(\mathcal{F}_n\) be the \(\sigma\)-algebra defined as
\[
\mathcal{F}_n=\sigma\{[kP^{-1}_n,(k+1)P^{-1}_n):0\leq k\leq P_n\}.
\]
Therefore, the interval of the form \([kP^{-1}_n,(k+1)P^{-1}_n)\) for some \(k,n\in\mathbb{N}\) and \(0\leq k\leq P_n\), is called a Vilenkin interval. An integrable sequence \(f=(f_n)_{n\in\mathbb{N}}\) is a martingale if \(f_n\) is \(\mathcal{F}_n\)-measurable \(\forall n\in\mathbb{N}\) and \(E_nf_m=f_n\) if \(n\leq m\). Hence, a martingale with respect to \((\mathcal{F}_n)_n\) is called a Vilenkin martingale. In this way, the author introduces the maximal function given by \(M(f)=\sup_{n\in\mathbb{N}}|f_n|\) and the variable martingale Hardy spaces are defined as
\[
H_{p(.)}=\{f=(f_n)_{n\in\mathbb{N}}:\|f\|_{H_{p(.)}}=\|M(f)\|_{p(.)}<\infty\},
\]
and
\[
H_{p(.),q}=\{f=(f_n)_{n\in\mathbb{N}}:\|f\|_{H_{p(.),q}}=\|M(f)\|_{p(.),q}<\infty\}.
\]
Now, every point \(x\in[0,1)\) can be written as \(x=\sum_{k=0}^\infty \frac{x_k}{P_{k+1}}\), \(0\leq x_k\leq p_k\), \(x_k\in \mathbb{N}\) and the functions \(r_n(x)=\exp(2\pi i/p_n)\), \(n\in \mathbb{N}\), are called generalized Rademacher functions. Therefore, the functions \(w_n(x)=\prod_{k=0}^\infty r_k(x)^{n_k}\) constitute a Vilenkin system, where \(n=\sum_{k=0}^{\infty}n_kP_k\) and \(0\leq n_k\leq p_k\). The Vilenkin Dirichlet kernels are defined as \(D_n=\sum_{k=0}^{n-1}w_k\). Also, if \(f\in L_1\) then \(\hat{f}(n)=\int_0^1 f\bar{w_n}d\lambda\), \(n\in\mathbb{N}\), is the Vilenkin-Fourier coefficient and if \(f=(f_n)_{n\in\mathbb{N}}\) is a martingale, then \(\hat{f}(n)=\lim_{k\to\infty}\int_0^1 f_k\bar{w_n}d\lambda\). The \(n\)th partial sum of the Vilenkin Fourier series of a martingale \(f\) is given as
\[
S_nf=\sum_{k=0}^{\infty}\hat{f}(k)w_k.
\]
In this context, the author obtains the following result. Let \(p(.)\in C^{\log}\) with \(1<p\_\leq p_{+}<\infty\). If \(f\in L_{p(.)}\) then \(\sup_{n\in\mathbb{N}}\|S_n f\|_{p(.)}\lesssim \|f\|_{p(.)}\). Moreover, \(\lim_{n\to\infty}S_nf=f\) in the \(L_{p(.)}\)-norm.
Finally, let us consider the Fejer mean of order \(n\) of the Vilenkin-Fourier series of \(f\), as \(\sigma_nf=\frac{1}{n}\sum_{k=1}^{n}S_kf\) and the maximal operators \(\sigma_*f=\operatorname*{sup}_{n\in\mathbb{N}}|\sigma_nf|\). Then, through atomic decompositions, the author gets the following results, which we summarize as follows:
Theorem 1.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\). If \(\frac{1}{2}<p\_<\infty\), then \(\| \sigma_*f\|_{p(.)}\lesssim \|f\|_{H_{p(.)}}\), for \(f\in H_{p(.)}\).
Theorem 2.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\). If \(0<q\leq \infty\), then
\[
\|\operatorname*{sup}_{n\in\mathbb{N}}|\sigma_{P_n}f| \|_{L_{p(.)},q}\lesssim \|f\|_{H_{p(.)},q},
\ \text{for}
\ f\in H_{p(.),q}
\]
and \(\| \sigma_*f\|_{p(.),q}\lesssim \|f\|_{H_{p(.)},q}\), for \(f\in H_{p(.),q}\).
Corollary 1.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\) and \(0<q\leq \infty\). If \(f\in H_{p(.),q}\) then \(\sigma_{P_n}f\) converges almost everywhere on \([0,1)\) and in the \(L_{p(.),q}\) norm. If in addition, \(\frac{1}{2}<p\_<\infty\), then the convergence holds for \(\sigma_n f\) too.
Corollary 2.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\), \(1\leq p\_<\infty\), \(0<q\leq \infty\) and \(f\in H_{p(.),q}\). Then, \(\lim_{n\to\infty} \sigma_nf(x)=f(x)\), for a.e. \(x\in[0,1)\) and in the \(L_{p(.),q}\) norm.
Corollary 3.
If \(f\in L_1\), then \(\lim_{n\to\infty}\sigma_nf(x)=f(x)\) for a.e. \(x\in [0,1)\).
Reviewer: Iris Athamaica López Palacios (Caracas)Time-frequency analysis associated with \(k\)-Hankel-Wigner transformshttps://zbmath.org/1526.420152024-02-15T19:53:11.284213Z"Boubatra, Mohamed Amine"https://zbmath.org/authors/?q=ai:boubatra.mohamed-amineSummary: In this paper, we introduce the \(k\)-Hankel-Wigner transform on \(\mathbb{R}\) in some problems of time-frequency analysis. As a first point, we present some harmonic analysis results such as Plancherel's, Parseval's and an inversion formulas for this transform. Next, we prove a Heisenberg's uncertainty principle and a Calderón's reproducing formula for this transform. We conclude this paper by studying an extremal function for this transform.Expanding the Fourier transform of the scaled circular Jacobi \(\beta\) ensemble densityhttps://zbmath.org/1526.420162024-02-15T19:53:11.284213Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Shen, Bo-Jian"https://zbmath.org/authors/?q=ai:shen.bo-jianSummary: The family of circular Jacobi \(\beta\) ensembles has a singularity of a type associated with \textit{M. E. Fisher} and \textit{R. E. Hartwig} [Adv. Chem. Phys. 15, 333--353 (1968; \url{doi.org/10.1002/9780470143605.ch18})] in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding \(N \rightarrow \infty\) bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi \(\beta\) ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for \(\beta = 2\) and \(\beta = 4\), and the loop equation hierarchy. The polynomials in the variable \(u=2/\beta\) which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular \(\beta\) ensemble, specifically in relation to the zeros lying on the unit circle \(|u|=1\) and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.Global Gevrey vectorshttps://zbmath.org/1526.420172024-02-15T19:53:11.284213Z"Hoepfner, G."https://zbmath.org/authors/?q=ai:hoepfner.gustavo"Raich, A."https://zbmath.org/authors/?q=ai:raich.andrew-s"Rampazo, P."https://zbmath.org/authors/?q=ai:rampazo.pSummary: In this paper, we introduce the notion of global \(L^q\) Gevrey vectors and investigate the regularity of such vectors in global and microglobal settings when \(q=2\). We characterize the vectors in terms of the FBI transform and prove global and microglobal versions of the Kotake-Narasimhan Theorem. Our techniques are new because our results are written in terms of the FBI transform and not the Fourier transform. Additionally, the microglobal Kotake-Narasimhan Theorem provides a refinement of an earlier result by \textit{G. Hoepfner} and \textit{A. Raich} [Math. Z. 291, No. 3--4, 971--998 (2019; Zbl 1416.42008)] relating the microglobal wavefront sets of the ultradistributions \(u\) and \(Pu\) when \(P\) is a constant coefficient differential operator.Dimensional estimates for measures on quaternionic sphereshttps://zbmath.org/1526.420182024-02-15T19:53:11.284213Z"Ayoush, Rami"https://zbmath.org/authors/?q=ai:ayoush.rami"Wojciechowski, Michał"https://zbmath.org/authors/?q=ai:wojciechowski.michalSummary: In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansions. This estimate is an analog of a result previously obtained by the authors for the complex spheres.Endpoint sparse domination for classes of multiplier transformationshttps://zbmath.org/1526.420192024-02-15T19:53:11.284213Z"Beltran, David"https://zbmath.org/authors/?q=ai:beltran.david"Roos, Joris"https://zbmath.org/authors/?q=ai:roos.joris"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreasSummary: We prove endpoint results for sparse domination of translation invariant multiscale operators. The results are formulated in terms of dilation invariant classes of Fourier multipliers based on natural localized \(M^{p \to q}\) norms which express appropriate endpoint regularity hypotheses. The applications include new and optimal sparse bounds for classical oscillatory multipliers and multi-scale versions of radial bump multipliers.Singular integral operators with rough kernel on function spaces over local fieldshttps://zbmath.org/1526.420202024-02-15T19:53:11.284213Z"Ashraf, Salman"https://zbmath.org/authors/?q=ai:ashraf.salman"Jahan, Qaiser"https://zbmath.org/authors/?q=ai:jahan.qaiserSummary: In this article we study the boundedness of classical singular integral operator over local fields with rough kernel. We are relaxing the smoothness condition on kernel by block spaces and found the boundedness of truncated singular integral operator on different function spaces such as Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Unlike Euclidean space, our bound of truncated singular integral operator depends on the constant which leads the unboundedness of singular integral operator over local fields.Uniform sparse domination and quantitative weighted boundedness for singular integrals and application to the dissipative quasi-geostrophic equationhttps://zbmath.org/1526.420212024-02-15T19:53:11.284213Z"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.3|chen.yanping.2|chen.yanping.1"Guo, Zihua"https://zbmath.org/authors/?q=ai:guo.zihuaSummary: In this paper, we consider a kind of singular integrals
\[
T_\lambda f (x) = \text{ p.v. } \int\limits_{\mathbb{R}^n} \frac{\Omega (y)}{|y|^{n - \lambda}} f (x - y) dy
\]
for any \(f \in L^q (\mathbb{R}^n)\), \(1 < q < \infty\) and \(0 < \lambda < n\), which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation
\[
\partial_t \theta + u \cdot \nabla \theta + \kappa \Lambda^{2 \beta} \theta = 0, \quad (x, t) \in \mathbb{R}^2 \times \mathbb{R}^+,\ \kappa > 0,
\tag{0.1}
\]
where \(u = - \nabla^\bot \Lambda^{- 2 + 2 \alpha} \theta\), \(\alpha \in [0, \frac{1}{2}]\) and \(\beta \in (0, 1]\). Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator \(T_\lambda\) with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator \([b, T_\lambda]\) with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation.Commutators of multilinear Calderón-Zygmund operators with kernels of Dini's type on generalized weighted Morrey spaces and applicationshttps://zbmath.org/1526.420222024-02-15T19:53:11.284213Z"Guliyev, V. S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabirThe paper establishes weighted Morrey strong and weak \(L(\log L)\)-type endpoint estimates for multilinear Calderón-Zygmund operators with kernels of Dini type, and for iterated commutators of such operators with functions in BMO\(^m\). The results are applied to give weighted Morrey estimates for iterated commutators of bilinear pseudo-differential operators and paraproducts with mild regularity.
Reviewer: Guorong Hu (Nanchang)Some remarks on convex body dominationhttps://zbmath.org/1526.420232024-02-15T19:53:11.284213Z"Hytönen, Tuomas P."https://zbmath.org/authors/?q=ai:hytonen.tuomas-pSummary: Convex body domination is an important elaboration of the technique of sparse domination that has seen significant development and applications over the past ten years. In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued functions, and yields matrix-weighted norm inequalities in this setting. We explore applications to ``generalised commutators'', obtaining new examples of bounded operators among linear combinations of compositions of the form \(a_i T b_i\), where \(a_i\), \(b_i\) are pointwise multipliers and \(T\) is a singular integral operator.Parabolic non-singular integral operator and its commutators on parabolic vanishing generalized Orlicz-Morrey spaceshttps://zbmath.org/1526.420242024-02-15T19:53:11.284213Z"Omarova, M. N."https://zbmath.org/authors/?q=ai:omarova.mehriban-n|omarova.meriban-nSummary: We obtain the sufficient conditions for the boundedness of the parabolic nonsingular integral operator and its commutators on the parabolic vanishing generalized Orlicz Morrey spaces \(M^{\Phi,\varphi}(\mathbb{D}^{n+1}_+)\) including their weak versions.Bloom type inequality: the off-diagonal casehttps://zbmath.org/1526.420252024-02-15T19:53:11.284213Z"Pan, Junren"https://zbmath.org/authors/?q=ai:pan.junren"Sun, Wenchang"https://zbmath.org/authors/?q=ai:sun.wenchangSummary: In this paper, we establish a representation formula for fractional integrals. As a consequence, we get a Bloom type inequality for the Ferguson-Lacey type commutator involved with fractional integrals. Our results extend similar results for singular integral operators. The main difference is that mixed-norm spaces are invoked when we study the off-diagonal case of Bloom type inequalities for fractional integral operators. Specifically, for two fractional integral operators \(I_{\lambda_1}\) and \(I_{\lambda_2}\), we prove a Bloom type inequality
\[
\begin{gathered}
\left\| \left[I^1_{\lambda_1}, \left[b, I^2_{\lambda_2}\right]\right]\right\|_{L^{p_2} (L^{p_1})(\mu_2^{p_2}\times \mu_1^{p_1})\to L^{q_2} (L^{q_1}) (\sigma^{q_2}_2 \times \sigma_1^{q_1})} \\
\lesssim_{\substack{[\mu_1]_{A_{p_1,q_1} (\mathbb{R}^n)}, [\mu_2]_{A_{p_2,q_2} (\mathbb{R}^m)} \\ [\sigma_1]_{A_{p_1,q_1} (\mathbb{R}^n)}, [\sigma_2]_{A_{p_2,q_2} (\mathbb{R}^m)}}} \| b\|_{\mathrm{BMO}_{\mathrm{prod}}} (\nu),
\end{gathered}
\]
where the indices satisfy \(1<p_1 <q_1 <\infty\), \(1<p_2 <q_2 <\infty\), \(1/q_1 +1/p_1^{\prime}=\lambda_1 /n\) and \(1/q_2 +1/p_2^{\prime} =\lambda_2 /m\), the weights \(\mu_1,\sigma_1 \in A_{p_1,q_1}(\mathbb{R}^n)\), \(\mu_2,\sigma_2 \in A_{p_2,q_2}(\mathbb{R}^m)\) and \(\nu :=\mu_1\sigma_1^{-1}\otimes \mu_2\sigma_2^{-1}\), \(I_{\lambda_1}^1\) stands for \(I_{\lambda_1}\) acting on the first variable and \(I_{\lambda_2}^2\) stands for \(I_{\lambda_2}\) acting on the second variable, \(\mathrm{BMO}_{\mathrm{prod}}(\nu)\) is a weighted product \(\mathrm{BMO}\) space and \(L^{p_2}(L^{p_1})(\mu_2^{p_2}\times \mu_1^{p_1})\) and \(L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times \sigma_1^{q_1})\) are mixed-norm spaces.A \(T(P)\)-theorem for Zygmund spaces on domainshttps://zbmath.org/1526.420262024-02-15T19:53:11.284213Z"Vasin, A. V."https://zbmath.org/authors/?q=ai:vasin.andrei-v"Dubtsov, E. S."https://zbmath.org/authors/?q=ai:doubtsov.evgueniSummary: Given a bounded Lipschitz domain \(D\subset \mathbb{R}^d\), a higher-order modulus of continuity \(\omega \), and a convolution Calderón-Zygmund operator \(T\), the restricted operators \(T_D\) that are bounded on the Zygmund space \(\mathcal{C}_{\omega}(D)\) are described. The description is based on properties of the functions \(T_D P\) for appropriate polynomials \(P\) restricted to \(D\).The necessity theory for commutators of multilinear singular integral operators: the weighted casehttps://zbmath.org/1526.420272024-02-15T19:53:11.284213Z"Wang, Dinghuai"https://zbmath.org/authors/?q=ai:wang.dinghuaiSummary: In this paper, the necessity of BMO for boundedness of commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction on the weight class to general multiple weights, which can be regarded as an essential improvement of the result of \textit{L. Chaffee} and \textit{D. Cruz-Uribe} [Math. Inequal. Appl. 21, No. 1, 1--16 (2018; Zbl 1384.42012)] and \textit{W. Guo} et al. [J. Geom. Anal. 30, No. 4, 3995--4035 (2020; Zbl 1475.42024)]. Our approach elaborates on a commonly used method of expanding the kernel locally in Fourier series, recovering many known results but yielding also numerous new ones. In particular, we answer the question about the necessity issue for iterated commutators of multilinear singular integral operators.On the boundedness of rough bi-parameter Fourier integral operatorshttps://zbmath.org/1526.420282024-02-15T19:53:11.284213Z"Wang, Guangqing"https://zbmath.org/authors/?q=ai:wang.guangqing"Li, Jinhui"https://zbmath.org/authors/?q=ai:li.jinhuiSummary: Let the bi-parameter Fourier integral operators be defined by the phase functions \(\varphi_1(x_1,\xi)\), \(\varphi_2(x_2,\xi)\in L^\infty \Phi^2\) satisfying the rough non-degeneracy condition and the amplitude \(a\in L^pBS^m_{\varrho}\) with \(m=(m_1,m_2)\in{\mathbb{R}}^2\), \(\varrho =(\varrho_1,\varrho_2)\in [0,1]\times [0,1]\). It is proved that if \(0<r\le \infty\), \(1\le p\), \(q\le \infty\), satisfying the relation \(\frac{1}{r}=\frac{1}{q}+\frac{1}{p}\), then these operators are bounded from \(L^q\) to \(L^r\) provided
\[
m_i<-\varrho_i\frac{(n-1)}{2}\Bigg (\frac{1}{s}+\frac{1}{\min (p,s^\prime)}\Bigg)+\frac{n(\varrho_i-1)}{s} \quad i=1,2,
\]
where \(s=\min (2,p,q)\) and \(\frac{1}{s}+\frac{1}{s^\prime}=1\).Sharp weak-type estimates for maximal operators associated to rare baseshttps://zbmath.org/1526.420292024-02-15T19:53:11.284213Z"Hagelstein, Paul"https://zbmath.org/authors/?q=ai:hagelstein.paul-alton"Oniani, Giorgi"https://zbmath.org/authors/?q=ai:oniani.giorgi-gigla|oniani.giorgi"Stokolos, Alex"https://zbmath.org/authors/?q=ai:stokolos.alexThe authors consider a geometric maximal operator \(M_{\mathcal{B}}\) associated with a translation invariant collection \(\mathcal{B}\) of intervals in \(\mathbb{R}^n\), i.e.
\[
M_{\mathcal{B}} f(x) = \sup_{ R \in \mathcal{B} : x \in R } \frac{1}{|R|} \int_R |f| .
\]
Such a collection \(\mathcal{B}\) is called a rare basis. An interval in \(\mathbb{R}^n\) is a rectangular parallelepiped whose sides are parallel to the coordinate axes. Then, sufficient conditions on a rare basis \(\mathcal{B}\) are given so that the weak-type \(L(1+ \log^+ L)^{n-1}\) estimate
\[
\tag{1} | \{ x \in \mathbb{R}^n : M_{\mathcal{B}} f(x) > \alpha \} | \leq C_n \int_{\mathbb{R}^n} \frac{|f|}{\alpha} \bigg( 1 + \log^+ \frac{|f|}{\alpha} \bigg)^{n-1}
\]
is sharp for \(M_{\mathcal{B}}\). Subsequently, several applications on sharp weak-type \(L(1+ \log^+ L)^{n-1}\) estimates are provided for maximal operators associated to several classes of rare bases.
Specifically, the spectrum of \(\mathcal{B}\), denoted by \(W_{\mathcal{B}}\), is defined as the set of all \(n\)-tuples of the type \( ( \lceil \log_2 |R_1| \rceil , \ldots , \lceil \log_2 |R_n| \rceil ) \), where \(R_1 \times \ldots \times R_n \in \mathcal{B}\) and \(\lceil x \rceil\) denotes the least integer greater than or equal to \(x\).
A set \(W \subset \mathbb{Z}\) is called a net for a set \(S \subset \mathbb{Z}\) if there exists a natural number \(N\) such that for every \(s \in S\) there exists \(w \in W\) with \( |s-w| \leq N \).
Given a set \(W \subset \mathbb{Z}^n\) and \(t \in \mathbb{Z}^{n-1}\), denote \(W_t = \{ \tau \in \mathbb{Z} : (t, \tau) \in W \}\). Then, a set \(W \subset \mathbb{Z}^n\) is called a net for a set \(S \subset \mathbb{Z}^n\) if \(W_t\) is a net for \(S_t\) for every \(t \in \mathbb{Z}^{n-1}\).
Let \(\pi_k\) denote the usual projection \( \pi_k (x_1, \ldots , x_n) = (x_1, \ldots ,x_k) \). So, a set \(W \subset \mathbb{Z}^n\) is said to be dense in a set \(S_1 \times \ldots \times S_n \subset \mathbb{Z}^n\) if the sets \(\pi_1 (W), \ldots, \pi_n (W)\) are nets for the sets \( S_1, \pi_1 (W) \times S_2, \ldots , \pi_{n-1} (W) \times S_n \) respectively.
The main theorem of this article is the following.
Theorem. If for a rare basis \(\mathcal{B}\), there exist infinite sets \(S_1, \ldots , S_n \subset \mathbb{Z}\) for which the spectrum of \(\mathcal{B}\) is dense in \(S_1 \times \ldots \times S_n\), then the maximal operator \(M_{\mathcal{B}}\) satisfies a sharp weak-type \(L(1+ \log^+ L)^{n-1}\) estimate as in (1). Moreover, for every \(\alpha \in (0,1)\), there exists a bounded set \(E_\alpha \subset \mathbb{R}^n\) with positive measure such that
\[
| \{ x \in \mathbb{R}^n : M_{\mathcal{B}} (\chi_{E_\alpha}) (x) > \alpha \} | \geq c_n \frac{1}{\alpha} \bigg( 1 + \log \frac{1}{\alpha} \bigg)^{n-1} |E_\alpha| .
\]
Geometric maximal operators associated with rare bases occupy a fascinating middle ground between the Hardy-Littlewood maximal operator (the basis \(\mathcal{B}\) consists of all cubic intervals) and the strong maximal operator (the basis \(\mathcal{B}\) consists of all intervals). Significant mathematical work on the topic on rare bases has been done by, among others, Zygmund, Córdoba, Soria, and Rey.
Reviewer: Guillermo Flores (Córdoba)Regularity of general maximal and minimal functionshttps://zbmath.org/1526.420302024-02-15T19:53:11.284213Z"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.192"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng.14|liu.feng.1|liu.feng|liu.feng.2|liu.feng.4|liu.feng.5Summary: In this paper, our object of investigation is the endpoint regularity of the following general maximal operator,
\[
\widetilde{\mathcal{M}}_\Phi f(x) = \sup\limits_{\substack{r, s \geq 0\\ r+s>0}}\Phi (r+s)\int_{x-r}^{x+s}|f(y)|\mathrm{d}y,
\]
and minimal operator,
\[
\widetilde{m}_\Phi f(x) = \inf\limits_{\substack{r, s \geq 0\\ r+s>0}}\Phi (r+s)\int_{x-r}^{x+s}|f(y)|\mathrm{d}y,
\]
where \(\Phi(t): (0, \infty)\rightarrow(0, \infty)\) is a non-increasing continuous function and satisfies \(B_q := \sup_{t > 0}t\Phi (t)^q < \infty\) for some \(q \geq 1\). We prove that if \(f:\mathbb{R}\rightarrow \mathbb{R}\) is a function of bounded variation, then
\[
\max\{\mathrm{Var}_q(\widetilde{\mathcal{M}}_\Phi f), \mathrm{Var}_q(\widetilde{m}_\Phi f)\} \leq (8B_q)^{1/q}\mathrm{Var}(f).
\]
Here, \(\mathrm{Var}_q(f)\) denotes the \(q\)-variation of \(f\) and \(\mathrm{Var}_q(f) = \mathrm{Var}(f)\) when \(q = 1\). Similar results are proved for the discrete versions of the above operators.Some quantitative one-sided weighted estimateshttps://zbmath.org/1526.420312024-02-15T19:53:11.284213Z"Lorente, María"https://zbmath.org/authors/?q=ai:lorente.maria"Martín-Reyes, Francisco J."https://zbmath.org/authors/?q=ai:martin-reyes.francisco-javier"Rivera-Ríos, Israel P."https://zbmath.org/authors/?q=ai:rivera-rios.israel-pSummary: In this paper we provide some quantitative one-sided estimates that recover the dependences in the classical setting. Among them we provide estimates for the one-sided maximal function in Lorentz spaces and we show that the conjugation method for commutators works as well in this setting.Remarks on vector-valued Gagliardo and Poincaré-Sobolev-type inequalities with weightshttps://zbmath.org/1526.420322024-02-15T19:53:11.284213Z"Perales, Javier Martínez"https://zbmath.org/authors/?q=ai:perales.javier-martinez"Pérez, Carlos"https://zbmath.org/authors/?q=ai:perez.carlos-javier|perez.carlos-e|perez-moreno.carlos|perez.carlos-aSummary: In this paper, we review certain extensions of the Gagliardo and Poincaré- Sobolev-type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical case of \(\ell_q\)-valued functions, where already some difficulties arise, due to the lack of a vector-valued variant of the truncation method, both on the classical and the fractional case. We think that these difficulties may be overcome in the future, and we pose some conjectures in this direction.
For the entire collection see [Zbl 1515.31001].Spherical maximal operators on Heisenberg groups: restricted dilation setshttps://zbmath.org/1526.420332024-02-15T19:53:11.284213Z"Roos, Joris"https://zbmath.org/authors/?q=ai:roos.joris"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Srivastava, Rajula"https://zbmath.org/authors/?q=ai:srivastava.rajulaSummary: Consider spherical means on the Heisenberg group with a codimension 2 incidence relation, and associated spherical local maximal functions \(M_E f\) where the dilations are restricted to a set \(E\). We prove \(L^p \to L^q\) estimates for these maximal operators; the results depend on various notions of dimension of \(E\).On the boundedness of the maximal operators associated with singular hypersurfaceshttps://zbmath.org/1526.420342024-02-15T19:53:11.284213Z"Usmanov, S. E."https://zbmath.org/authors/?q=ai:usmanov.s-eSummary: The paper deals with maximal operators associated with a family of singular hypersurfaces in the space \(\mathbb{R}^{n+1} \). The boundedness of these operators in the space of integrable functions is proved for the case in which the singular hypersurfaces are given by parametric equations. The boundedness exponent of maximal operators for spaces of integrable functions is found.A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaceshttps://zbmath.org/1526.420352024-02-15T19:53:11.284213Z"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiro"Kobayashi, Kazuki"https://zbmath.org/authors/?q=ai:kobayashi.kazukiSummary: The purpose of the present note is to slightly shorten the proof of the atomic decomposition based on the paper by \textit{S. Dekel} et al. [in: Constructive theory of functions. Proceedings of the 12th international conference, Sozopol, Bulgaria, June 11--17, 2016. Sofia: Prof. Marin Drinov Academic Publishing House. 59--73 (2018; Zbl 1447.42020)]. The atomic decomposition in the present paper is applicable to Hardy spaces based on the convexification of ball Banach spaces. The decomposition is rather canonical although it does not depend linearly on functions. Also, this decomposition is applicable under a rather weak condition as we will see.
For the entire collection see [Zbl 1515.31001].Dyadic lower little BMO estimateshttps://zbmath.org/1526.420362024-02-15T19:53:11.284213Z"Domelevo, K."https://zbmath.org/authors/?q=ai:domelevo.komla"Kakaroumpas, S."https://zbmath.org/authors/?q=ai:kakaroumpas.spyridon"Petermichl, S."https://zbmath.org/authors/?q=ai:petermichl.stefanie"Soler i Gibert, Odí"https://zbmath.org/authors/?q=ai:soler-i-gibert.odiSummary: We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well-chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case and with Bloom weights. Moreover, we address the flexibility of one of our methods.Duality of three-parameter Hardy spaces associated with a sum of two flag singular integralshttps://zbmath.org/1526.420372024-02-15T19:53:11.284213Z"He, Shaoyong"https://zbmath.org/authors/?q=ai:he.shaoyong"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiechengSummary: We establish the duality theory of the three-parameter Hardy spaces associated with a sum of two flag singular integrals and characterize this dual space as the sum of flag Carleson measure spaces. Our work requires more complicated analysis associated with the underlying geometry generated by the multi-parameter structures.Extrapolation in new weighted grand Morrey spaces beyond the Muckenhoupt classeshttps://zbmath.org/1526.420382024-02-15T19:53:11.284213Z"Meskhi, Alexander"https://zbmath.org/authors/?q=ai:meskhi.alexanderSummary: Rubio de Francia's extrapolation theorem for new weighted grand Morrey spaces \(\mathcal{M}_w^{p), \lambda, \theta}(X)\) with weights \(w\) beyond the Muckenhoupt \(\mathcal{A}_p\) classes is established. This result, in particular, implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. Necessary conditions for the boundedness of the Hardy-Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are also investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applicationshttps://zbmath.org/1526.420392024-02-15T19:53:11.284213Z"Zhao, Yichun"https://zbmath.org/authors/?q=ai:zhao.yichun"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangIn this paper, the authors introduce anisotropic mixed-norm Herz spaces and investigate some properties of those spaces. Furthermore, they establish the Rubio de Francia extrapolation theory, which is used to prove the boundedness of Calderón-Zygmund operators, fractional integral operators, and commutators on anisotropic mixed-norm Herz spaces. They also prove the Littlewood-Paley characterization by the \(g\)-function of anisotropic mixed-norm Herz spaces. Moreover, they introduce anisotropic mixed-norm Herz-Hardy spaces, on which atomic and molecular decompositions are obtained and they prove the boundedness of Calderón-Zygmund operators.
Reviewer: Koichi Saka (Akita)Approximation via gradients on the ball. The Zernike casehttps://zbmath.org/1526.420402024-02-15T19:53:11.284213Z"Marriaga, Misael E."https://zbmath.org/authors/?q=ai:marriaga.misael-e"Pérez, Teresa E."https://zbmath.org/authors/?q=ai:perez.teresa-e"Piñar, Miguel A."https://zbmath.org/authors/?q=ai:pinar.miguel-a"Recarte, Marlon J."https://zbmath.org/authors/?q=ai:recarte.marlon-jThe authors consider the inner product
\[
\langle f,g\rangle_{\nabla,\mu}=f(0)g(0)+\lambda\int_{\mathcal{B}^d}\nabla f(x)\cdot\nabla g(x) W_{\mu}(x)dx
\]
on the unit ball \( \mathcal{B}^d \) in \( \mathbb{R}^d \) with normalizing factor \( \lambda>0 \) and \( W_{\mu}(x)=(1-\Vert x\Vert^2)^{\mu}\), \(\mu\geq 0 \). They determine an explicit orthogonal polynomial basis and deduce relations between the partial Fourier sums in terms of those polynomials and the partial Fourier sums in terms of the classical ball polynomials. Approximation properties of the Fourier series in the corresponding Sobolev space are studied, and numerical examples are given.
Reviewer: Alexei Lukashov (Moskva)Inequalities on the closeness of two vectors in a Parseval framehttps://zbmath.org/1526.420412024-02-15T19:53:11.284213Z"Ashino, Ryuichi"https://zbmath.org/authors/?q=ai:ashino.ryuichi"Mandai, Takeshi"https://zbmath.org/authors/?q=ai:mandai.takeshi"Morimoto, Akira"https://zbmath.org/authors/?q=ai:morimoto.akiraLet \(H\) be a Hilbert space and let \(K\) be an index set. A sequence \(F=\{f_k\}_{k\in K}\) in \(H\) is called a frame if there exist two positive constants \(A\), \(B\) such that
\[
A\|f\|^2\leq\sum_{k\in K}|\langle f,f_k\rangle|^2\leq B\|f\|^2,
\]
for each \(f\in H\). The sequence \(F\) is called Parseval if \(A=B=1\).
In the present paper, some properties of frames in Hilbert spaces are obtained, especially Parseval frames are considered and studied.
It is easy to see that if \(F=\{f_k\}_{k\in K}\) is a Parseval frame and \(\|f_{k_0}\|=1\), for some \(k_0\in K\), then \(f_{k_0}\perp f_k\), for every \(k\neq k_0\) (so it is expected that if \(\|f_{k_0}\|\) is near 1, then the angle between \(f_{k_0}\) and \(f_k\) (\(k\neq k_0\)) is near \(\frac{\pi}{2}\), that is \(f_{k_0}\) and \(f_k\) are not so close to each other). Using this fact, the main result of the paper, which is an inequality for Parseval frames, is stated as follows:
Let \(F=\{f_k\}_{k\in K}\) be a Parseval frame. If \(k\neq l\), then
\[
|\langle f_k,f_l\rangle|\leq\sqrt{1-\|f_k\|^2}\sqrt{1-\|f_l\|^2}.
\]
In other words, if \(f_k,f_l\neq 0\), then
\[
\frac{|\langle f_k,f_l\rangle|}{\|f_k\|\|f_l\|}\leq\frac{\sqrt{1-\|f_k\|^2}\sqrt{1-\|f_l\|^2}}{\|f_k\|\|f_l\|}.
\]
Then, the authors conclude that if \(\|f_k\|\) is near 1 and if \(\|f_l\|\) is not so small, then \(f_k\) and \(f_l\) are not so close to each other.
Reviewer: Morteza Mirzaee Azandaryani (Qom)Gabor frame bound optimizationshttps://zbmath.org/1526.420422024-02-15T19:53:11.284213Z"Faulhuber, Markus"https://zbmath.org/authors/?q=ai:faulhuber.markus"Shafkulovska, Irina"https://zbmath.org/authors/?q=ai:shafkulovska.irinaThe authors find extremal lattices for the spectral bounds of Gabor systems with specific windows. The quantities that they optimize are the lower and upper frame bounds as well as their ratio, which is the condition number of the associated frame operator. They study the cases provided by \textit{A. J. E. M. Janssen} [Indag. Math., New Ser. 7, No. 2, 165--183 (1996; Zbl 1056.42512)], where sharp spectral bounds for Gabor frames over rectangular lattices of the form \(a Z \times b Z\) were computed for several different window functions and \((ab)^{-1} \in N\). Their results hold for rectangular lattices of integer density and for different windows.
This paper is organized as follows. Section 1 is introductory. The results are presented in Section 2. Section 3 settles the notation and provides some background information and motivation. Section 4 contains explanations of auxiliary techniques used in the manuscript. The proofs of the results follow in Section 5 for the hyperbolic secant, Section 6 for cut-off exponentials, Section 7 for one-sided exponentials, and Section 8 for two-sided exponentials.
The rigorous analytic study of optimal lattices for Gabor systems is relatively new. The first work on this topic is apparently due to \textit{M. Faulhuber} and \textit{S. Steinerberger} [J. Math. Anal. Appl. 445, No. 1, 407--422 (2017; Zbl 1351.42039)].
A number of the computations involved in this work are performed using software and are available for download as a supplement.
Reviewer: Richard A. Zalik (Auburn)Construction of infinite frames with some given redundancyhttps://zbmath.org/1526.420432024-02-15T19:53:11.284213Z"Hasankhani Fard, Mohammad Ali"https://zbmath.org/authors/?q=ai:fard.mohammad-ali-hasankhani\textit{B. G. Bodmann} et al. [Appl. Comput. Harmon. Anal. 30, No. 3, 348--362 (2011; Zbl 1211.42027)] introduced the notion of upper and lower redundancy for finite frames. These notions were generalized to infinite frames by \textit{J. Cahill} et al. [``A quantitative notion of redundancy for infinite frames'', Preprint, \url{arXiv:1006.2678}]. The paper under review makes a contribution to the study of redundancy for infinite frames. The first interesting result in the paper is an explicit construction of an infinite frame whose both lower and upper redundancy are arbitrarily close to one. Using norm one linear operators, a new formula for the upper redundancy of a frame is provided. It is proved in the paper that there is no frame having a lower redundancy of less than one and an upper redundancy of one. For certain parameters, frames are constructed having lower and upper redundancy given by those parameters.
Reviewer: Mahesh Krishna K. (Bangalore)Maltsev equal-norm tight frameshttps://zbmath.org/1526.420442024-02-15T19:53:11.284213Z"Novikov, Sergey Ya."https://zbmath.org/authors/?q=ai:novikov.sergey-ya"Sevost'yanova, Victoria V."https://zbmath.org/authors/?q=ai:sevostyanova.victoria-vSummary: A frame in \(\mathbb{R}^d\) is a set of \(n\geqslant d\) vectors whose linear span coincides with \(\mathbb{R}^d\). A frame is said to be equal-norm if the norms of all its vectors are equal. Tight frames enable one to represent vectors in \(\mathbb{R}^d\) in the form closest to the representation in an orthonormal basis. Every equal-norm tight frame is a useful tool for constructing efficient computational algorithms. The construction of such frames in \(\mathbb{C}^d\) uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in \(\mathbb{R}^d\) appeared only at the beginning of the 21st century. The present paper shows that Mal'tsev's note of 1947 [\textit{A. I. Mal'tsev}, Izv. Akad. Nauk SSSR, Ser. Mat. 11, 567--568 (1947; Zbl 0029.40502)] was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in \(\mathbb{R}^d\). Our main purpose is to show the historical significance of Maltsev's discovery. We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces. Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsev construction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence of any subset of \(d\) vectors in the frame.A class of structured frames in finite dimensional Hilbert spaceshttps://zbmath.org/1526.420452024-02-15T19:53:11.284213Z"Thomas, Jineesh"https://zbmath.org/authors/?q=ai:thomas.jineesh"Namboothiri, N. M. Madhavan"https://zbmath.org/authors/?q=ai:madhavan-namboothiri.n-m"Nambudiri, T. C. Easwaran"https://zbmath.org/authors/?q=ai:easwaran-nambudiri.t-cThe paper introduces the \(B\)-Gabor-like frames, which constitute a class of structured frames in the finite dimensional space \(\mathcal{H}\) generated by a single element. The symbol \(B\) indicates an invertible and bounded operator \(B:l^2(Z_\mathbb{N})\to \mathcal{H}\) which, together with the translation and the modulation operators, determines the structure of the frames. A characterisation of the frame operators associated with these frames is given. Also, the concept of Gabor semi-frames is introduced and some properties of the associated semi-frame operators are discussed.
Reviewer: Rosario Corso (Palermo)Existence of unconditional wavelet bases for \(L^p\)-norm over a local fields of positive characteristichttps://zbmath.org/1526.420462024-02-15T19:53:11.284213Z"Pathak, Ashish"https://zbmath.org/authors/?q=ai:pathak.ashish"Kumar, Dileep"https://zbmath.org/authors/?q=ai:kumar.dileepIn this paper, the authors define the Calderon-Zygmund operator on local fields of positive characteristics under various conditions on the kernel and then find the boundedness of that operator. Using a result on the boundedness of the Calderon-Zygmund operator and an orthonormal wavelet basis of local fields of positive characteristics, they provide unconditional wavelet bases for the \(L^p\)-norm by the wavelet coefficients over local fields of positive characteristic.
Reviewer: Mohd. Younus Bhat (Pulwama)Harmonic analysis on hypergroups. Approximation and stochastic sequenceshttps://zbmath.org/1526.430012024-02-15T19:53:11.284213Z"Lasser, Rupert"https://zbmath.org/authors/?q=ai:lasser.rupertHarmonic analysis is a broad notion. A concrete realization of certain basic features strongly depends on the setting where structural and analytic properties are investigated. In the book under review the setting is hypergroups, that is, sequences, functions, measures, functionals and operators are defined on hypergroups. The basic notion of translates is defined on hypergroups as a probability measure with compact support. One of the features of this book, where the number of pages exceeds 600, is that abstract harmonic analysis is combined with applied topics of spectral analysis and approximation by orthogonal expansions. The structure of the book is suitable. Before accounting the chapters and briefly their contents, we mention that the history of the subject is also given in detail.
Chapter 1 begins with preliminaries for defining a hypergroup as a locally compact Hausdorff space equipped with a convolution, an involution, and a unit element satisfying properties such as associativity or inversion. Also, in this chapter important classes of hypergroups are introduced. Chapter~2 is of preparatory nature as well. After the discussion of the left-invariant Haar measures and translation and convolution operators, representations of hypergroups are studied.
Chapters 3 and 4 are the most extensive in the book and form the essence of the subject studied. The former deals with commutative hypergroups, while the latter is devoted to the case where a hypergroup is generated by an orthogonal polynomial sequence. A variety of the corresponding properties are discussed in the both chapters and many analogs of classical results in other settings are established.
In the rest of the book, more specific properties of hypergroups are investigated. In a sense, these chapters may be considered as applications of the preceding material. In Chapter~5, weakly stationary random fields on commutative hypergroups are introduced and studied. In Chapter~6, weakly stationary random fields on polynomial hypergroups are studied. In Chapter~7, problems of difference equations of a special type are discussed. Finally, in Chapter~8, the author presents further hypergroups classes.
For those who search for a new setting in harmonic analysis (for some as a starting point), this book presents enough material and ideas to be absorbed in.
Reviewer: Elijah Liflyand (Ramat-Gan)Weak-type maximal function estimates on the infinite-dimensional torushttps://zbmath.org/1526.430052024-02-15T19:53:11.284213Z"Kosz, Dariusz"https://zbmath.org/authors/?q=ai:kosz.dariusz"Rey, Guillermo"https://zbmath.org/authors/?q=ai:rey.guillermo"Roncal, Luz"https://zbmath.org/authors/?q=ai:roncal.luzSummary: We prove necessary and sufficient conditions for the weak-\(L^p\) boundedness, for \(p \in (1,\infty)\), of a maximal operator on the infinite-dimensional torus. In the endpoint case \(p=1\) we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.Endpoint estimates and optimality for the generalized spherical maximal operator on radial functionshttps://zbmath.org/1526.440022024-02-15T19:53:11.284213Z"Nowak, Adam"https://zbmath.org/authors/?q=ai:nowak.adam"Roncal, Luz"https://zbmath.org/authors/?q=ai:roncal.luz"Szarek, Tomasz Z."https://zbmath.org/authors/?q=ai:szarek.tomasz-zacharySummary: We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions \(M^{ \alpha, \beta}_t\) to be bounded on power weighted Lebesgue spaces. Moreover, we also obtain the corresponding endpoint results in terms of optimal power weighted weak and restricted weak type estimates. All this complements significantly previous partial results existing in the literature.Anisotropic ball Campanato-type function spaces and their applicationshttps://zbmath.org/1526.460182024-02-15T19:53:11.284213Z"Li, Chaoan"https://zbmath.org/authors/?q=ai:li.chaoan"Yan, Xianjie"https://zbmath.org/authors/?q=ai:yan.xianjie"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunThe paper deals with the anisotropic ball Campanato-type function spaces associated with a dilation given by \(A\), a general expansive \(n\times n\) matrix, and \(X\), a ball quasi-Banach function space of \(\mathbb{R}^n\) (following the definition included in [\textit{Y.~Sawano} et al., Diss. Math. 525, 102~p. (2017; Zbl 1392.42021)]).
The authors show that these spaces are duals of anisotropic Hardy spaces \(H^{A}_X(\mathbb{R}^n)\), obtaining in this way anisotropic Littlewood-Paley function characterization of \(H^{A}_X(\mathbb{R}^n)\). Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both \(A\) and \(X\), further lead to their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-Slice spaces, some of the obtained results are new. Applications to the case of anisotropic Hardy-variable spaces are also included.
Reviewer: Santiago Boza (Barcelona)Real interpolation of variable martingale Hardy spaces and BMO spaceshttps://zbmath.org/1526.460192024-02-15T19:53:11.284213Z"Lu, Jianzhong"https://zbmath.org/authors/?q=ai:lu.jianzhong"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Zhou, Dejian"https://zbmath.org/authors/?q=ai:zhou.dejianSummary: In this paper, we mainly consider the real interpolation spaces for variable Lebesgue spaces defined by the decreasing rearrangement function and for the corresponding martingale Hardy spaces. Let \(0<q\leq \infty\) and \(0<\theta <1\). Our three main results are the following:
\[
\begin{aligned} ({\mathcal{L}}_{p(\cdot)}({\mathbb{R}}^n),L_{\infty}({\mathbb{R}}^n))_{\theta, q} &= {\mathcal{L}}_{{p(\cdot)}/(1-\theta),q}({\mathbb{R}}^n),\\
({\mathcal{H}}_{p(\cdot)}^s(\Omega),H_{\infty}^s(\Omega))_{\theta,q} &= {\mathcal{H}}_{{p(\cdot)}/(1-\theta),q}^s(\Omega) \end{aligned}
\]
and
\[
\begin{aligned} ({\mathcal{H}}_{p(\cdot)}^s(\Omega), \mathrm{BMO}_2(\Omega))_{\theta, q} = {\mathcal{H}}_{{p(\cdot)}/(1- \theta),q}^s(\Omega), \end{aligned}
\]
where the variable exponent \(p(\cdot)\) is a measurable function.Marcinkiewicz sampling theorem for Orlicz spaceshttps://zbmath.org/1526.460202024-02-15T19:53:11.284213Z"Pawlewicz, Aleksander"https://zbmath.org/authors/?q=ai:pawlewicz.aleksander"Wojciechowski, Michał"https://zbmath.org/authors/?q=ai:wojciechowski.michalSummary: In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an \(N\)-function \(\varphi\), boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterpartshttps://zbmath.org/1526.460232024-02-15T19:53:11.284213Z"Garrigós, Gustavo"https://zbmath.org/authors/?q=ai:garrigos.gustavo"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Ullrich, Tino"https://zbmath.org/authors/?q=ai:ullrich.tinoThe authors investigate the norm characterization for elements in Besov spaces \(B^s_{p,q}(\mathbb R)\) and Triebel-Lizorkin spaces \(F^s_{p,q}(\mathbb R)\) in terms of expressions involving their Haar coefficients or suitable variations thereof. The paper deals with the range of parameters \((s,p,q)\) in which the Haar system is not an unconditional basis. This complements previous work of the authors, e.g., [\textit{G.~Garrigós} et al., J. Geom. Anal. 31, No.~9, 9045--9089 (2021; Zbl 1478.46032)], where a complete description was given for the parameter range where the Haar system forms an unconditional basis or a Schauder basis.
The main aim of the paper under review is to see that the range of those characterizations previously shown in terms of Haar coefficients can be extended to other parameters provided that they doubly oversample with Haar coefficients obtained by a shift, namely, if \(h_{j,\mu}(x)= h(2^jx-\mu)\) for \(j=0,1,2, \dots\) and \(\mu\in \mathbb Z\), where \(h=\chi_{[0,1/2)}- \chi_{[1/2,1)}\) and \(h_{-1,\mu}=\chi_{[\mu,\mu+1)}\) stands for the Haar system in \(\mathbb R\), they consider \(\tilde h_{j,\nu}(x)= h(2^jx-\nu/2)\) for \(j=0,1,2, \dots\) and \(\nu\in \mathbb Z\) and \(\tilde h_{-1,\nu}=\chi_{[\nu,\nu+1)}\) and look at the extended Haar system \(\{\tilde h_{j,\nu}: h\ge -1,\,\nu\in \mathbb Z\}\). They use the notation \(c_{j,\mu}(f)= 2^j |\langle f, \tilde h_{j,2\mu}\rangle|+ 2^j |\langle f, \tilde h_{j,2\mu+1}\rangle|\) for \(j=0,1,2, \dots\) and \(c_{-1,\mu}(f)=\langle f, \chi_{[\mu,\mu+1)}\rangle\) and get characterizations of the norms in Besov and Triebel-Lizorkin spaces using these sequences, extending some previously known results which simply used the coefficients \(2^{j}\langle f, h_{j,\mu}\rangle\). They also show that, in the case \(1/p<s<1\), the classical Besov space \(B^s_{p,q}\) is a closed subset of its dyadic counterpart. They also provide equivalent norms for the Sobolev space \(W^1_p(\mathbb R)\) for \(1<p<\infty\) and some optimal inclusions between \(B^s_{p,q}\) and \(F^s_{p,q}\) and their dyadic counterparts.
Reviewer: Oscar Blasco (València)Dual spaces for martingale Musielak-Orlicz Lorentz Hardy spaceshttps://zbmath.org/1526.600332024-02-15T19:53:11.284213Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Xie, Guangheng"https://zbmath.org/authors/?q=ai:xie.guangheng"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunThis paper deals with various subspaces of martingales on a stochastic probability space with discrete time. A Musielak-Orlicz space on the underlying probability space generalizes the \(L^p\)-space by a more informative locally depending function instead of a global parameter \(p \ge 1\), and a Musielak-Orlicz Lorentz space controls further by a parameter \(q > 0\). Five martingale spaces are considered, which are called Martingale Musielak-Orlicz Lorentz Hardy spaces, depending on whether the maximal function, the quadratic variation and conditional quadratic variation, etc. of a martingale lies in the Musielak-Orlicz Lorentz space.
An atomic representation of such a martingale is one which is given by an infinite sum of special simpler functions. The authors show how, under some technical assumptions, these martingales of the martingale Musielak-Orlicz Lorentz Hardy spaces can be presented by such infinite sums of atoms. For the atomic presentations it is required that the sigma algebras of the filtration of the underlying probability space is generated by countably many atoms.
After some quasinorm estimates among the various martingale spaces, the paper continues with the introduction of martingale Musielak-Orlicz BMO-type spaces, and show how they are the dual spaces of martingale Musielak-Orlicz Lorentz Hardy spaces specified by conditional quadratic variation and maximal function, respectively.
Even if the paper appears somewhat technical the authors emphasize in a last discussion how the requirements can be easily checked in certain cases and how a vast class of similar martingale Hardy spaces known from the literature are covered by their paper.
The proof is by longer direct estimates and computations, and stopping times play an important role.
Reviewer: Bernhard Burgstaller (Florianópolis)QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn modelshttps://zbmath.org/1526.600492024-02-15T19:53:11.284213Z"Fernández, Lidia"https://zbmath.org/authors/?q=ai:fernandez.lidia"de la Iglesia, Manuel D."https://zbmath.org/authors/?q=ai:dominguez-de-la-iglesia.manuelSummary: We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.Asymptotic preserving and uniformly unconditionally stable finite difference schemes for kinetic transport equationshttps://zbmath.org/1526.650412024-02-15T19:53:11.284213Z"Zhang, Guoliang"https://zbmath.org/authors/?q=ai:zhang.guoliang"Zhu, Hongqiang"https://zbmath.org/authors/?q=ai:zhu.hongqiang"Xiong, Tao"https://zbmath.org/authors/?q=ai:xiong.tao.1Summary: In this paper, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We first derive an approximate evolution equation for the macroscopic density, from the formal solution of the distribution function, which is then discretized by following characteristics for the transport part with a backward finite difference semi-Lagrangian approach, while the diffusive part is discretized implicitly. After the macroscopic density is available, the distribution function can be efficiently solved even with a fully implicit time discretization since all discrete velocities are decoupled, resulting in a low-dimensional linear system from spatial discretizations at each discrete velocity. Both first and second order discretizations in space and in time are considered. The resulting schemes can be shown to be asymptotic preserving (AP) in the diffusive limit. Uniformly unconditional stabilities are verified from a Fourier analysis based on eigenvalues of corresponding amplification matrices. Numerical experiments, including high-dimensional problems, have demonstrated the corresponding orders of accuracy both in space and in time, uniform stability, AP property, and good performance of our proposed approach.Epicasting: an ensemble wavelet neural network for forecasting epidemicshttps://zbmath.org/1526.920552024-02-15T19:53:11.284213Z"Panja, Madhurima"https://zbmath.org/authors/?q=ai:panja.madhurima"Chakraborty, Tanujit"https://zbmath.org/authors/?q=ai:chakraborty.tanujit"Kumar, Uttam"https://zbmath.org/authors/?q=ai:kumar.uttam"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nanThis paper presents an ensemble wavelet neural network for forecasting epidemics, which does not have growing variance over time and exhibits good long-range forecast ability for epidemic datasets. The proposed formulation can handle nonlinear, non-stationary and long-range dependency of real-life data. Some theoretical properties, including asymptotic stationary, ergodicity, irreducibility and learning stability, have been examined. The paper explores the global characteristics of fifteen real-world infectious disease datasets including influenza, malaria, dengue, and hepatitis B from a range of areas. It is shown that the proposed framework can generate a better long term forecast and it outperforms many forecasters on average. A non-parametric test is used in the robustness study of the forecast.
Reviewer: Yilun Shang (Newcastle upon Tyne)Spectral inequalities for combinations of Hermite functions and null-controllability for evolution equations enjoying Gelfand-Shilov smoothing effectshttps://zbmath.org/1526.930132024-02-15T19:53:11.284213Z"Martin, Jérémy"https://zbmath.org/authors/?q=ai:martin.jeremy-l|martin.jeremy-m-r|martin.jeremy-g"Pravda-Starov, Karel"https://zbmath.org/authors/?q=ai:pravda-starov.karelSummary: This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index \(\frac{1}{2} \leq \mu <1\), we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand-Shilov space \(S^{\mu}_{\mu}(\mathbb{R}^n)\). These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.Information theory. Three theorems by Claude Shannonhttps://zbmath.org/1526.940012024-02-15T19:53:11.284213Z"Chambert-Loir, Antoine"https://zbmath.org/authors/?q=ai:chambert-loir.antoineThe fact that the amount of information can be numerically expressed and measured is now beyond dispute. Shannon's fundamental work entitled `A mathematical theory of communication' [\textit{C. E. Shannon}, Bell Syst. Tech. J. 27, 379--423, 623--656 (1948; Zbl 1154.94303)] played an important role in this.
This book aims to provide an introduction to information theory through three theorems of Shannon. Chapter 0. contains the most important basic concepts and results from probability theory required for the following chapters (summable families, probability theory, discrete random variables, independence and conditional expectation).
One of the main aims in the mathematical theory of communication is to analyse under which conditions a given channel may, or may not transmit a given message. Accordingly, the questions that must be answered are the following:
1. At what speed can one transmit a message?
2. Can one transmit a message in a reliable way in a noisy channel?
In order for these questions to be answered properly, it was first necessary to develop the concept that defines the amount of information contained in a given message. In modern terminology, this is nothing but the concept of entropy. Thus in Chapter 1, the author deals with the concepts of entropy and mutual information. At first, he recalls the notion of the entropy of a discrete random variable, that of condition entropy, mutual information, the entropy rate of a stochastic process, and also the entropy rate of Markov processes.
Chapter 2 focuses on coding. More precisely, the main problem studied here is how information can be transmitted in a noisy channel. Thus the author recalls the basics of this theory, such as alphabets, words, and codes. After that, with the aid of the Kraft-McMilman inequality, the following result of Shannon is presented.
Let \(X\) be a discrete random variable with values in a set \(A\). Let \(C\) be a code on an alphabet \(A\), with values in a finite alphabet \(B\) with cardinality \(D\geq 2\). If \(C\) is uniquely decodable, then the average length of \(C(X)\) satisfies the inequality
\[
\mathbb{E}\left(\ell(C(x))\right)\geq H_{D}(x),
\]
where \(H_{D}(X)\) is the base \(D\) entropy of \(X\). Equality holds if and only if \(\ell(C(a))= -\log_{D}(\mathbb{P}(X=a))\) for every \(a\in A\) such that \(\mathbb{P}(X=a)>0\).
This shows that entropy gives an unbreakable limit to the compression of a message. Therefore, later on, the author deals with how this limit can almost be reached, moreover by a prefix code. The following additional topics are discussed in the rest of the chapter: optimal codes, the law of large numbers, and compression, the transmission capacity of a channel, and coding adapted to a transmission channel.
Chapter 3 is about sampling theory. According to the sampling theorem, one may only retain samples of a given signal, taken at regularly spaced instants, provided the sampling frequency is at least twice that of the frequencies that `appear' in the signal. The discussion of sampling is impossible without the theory of Fourier series. Thus, the author first clarifies the most important concepts and statements, such as the Fourier series of periodic functions, the Bessel equation, the Parseval inequality, Dirichlet's theorem, and the Fourier transform. After all this, it is possible to present the most important results of the chapter, in Section 3.6 the sampling theorem, while in Section 3.7 the uncertainty principle in communication theory.
There are several exercises at the end of each chapter. A detailed solution to these can be found in the last, fourth chapter of the book. Since the author does not just write instructions and hints here, but detailed solutions can be found, this book can be especially useful for those who are just getting to know the basics of information theory.
Reviewer: Eszter Gselmann (Debrecen)Phase retrieval from intensity difference of linear canonical transformhttps://zbmath.org/1526.940082024-02-15T19:53:11.284213Z"Li, Youfa"https://zbmath.org/authors/?q=ai:li.youfa"Wu, Guangde"https://zbmath.org/authors/?q=ai:wu.guangde"Huang, Yanfen"https://zbmath.org/authors/?q=ai:huang.yanfen"Huang, Ganji"https://zbmath.org/authors/?q=ai:huang.ganjiSummary: The chirp-modulated shift-invariant space (CMSIS) is of interest in the linear canonical transform (LCT) based sampling theory. In this paper, we investigate the phase retrieval (PR) problem for the causal signals in CMSIS generated by sinc function. Unlike the traditional PR model, the measurement we use is the intensity difference of the LCT (IDLCT) but not the LCT intensity. The requirement for IDLCT measurement is weaker than that for intensity measurement. Such an IDLCT perspective is motivated by the famous PR model: the transport of intensity equation. It is revealed that the phase retrievability depends on both the LCT parameter matrix and the maximum gap of the target signal. From this, a necessary and sufficient condition for PR is established. Additionally, a sequence of sampling intervals are designed such that the PR can be achieved by the corresponding IDLCT measurements. Numerical simulations are conducted to check the correctness of the main results.Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditionshttps://zbmath.org/1526.940092024-02-15T19:53:11.284213Z"Wellershoff, Matthias"https://zbmath.org/authors/?q=ai:wellershoff.matthiasSummary: It was recently shown that functions in \(L^4 ([- B, B])\) can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transforms sampled on a rectangular lattice. We prove that this remains true if one replaces \(L^4 ([- B, B])\) by \(L^p ([- B, B])\) with \(p \in [1, \infty]\). To do so, we adapt the original proof by \textit{P. Grohs} and \textit{L. Liehr} [Appl. Comput. Harmon. Anal. 62, 173--193 (2023; Zbl 1504.94040)] and use a classical sampling result due to Beurling. Furthermore, we present a minor modification of a result of Müntz-Szász type by \textit{R. A. Zalik} [Trans. Am. Math. Soc. 243, 299--308 (1978; Zbl 0403.41008)]. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to \(L^p ([- B, B])\) and for more general nonuniform sampling sets.Recovery of rapidly decaying source terms from dynamical samples in evolution equationshttps://zbmath.org/1526.940152024-02-15T19:53:11.284213Z"Aldroubi, Akram"https://zbmath.org/authors/?q=ai:aldroubi.akram"Gong, Le"https://zbmath.org/authors/?q=ai:gong.le"Krishtal, Ilya"https://zbmath.org/authors/?q=ai:krishtal.ilya-arkadievichSummary: We analyze the problem of recovering a source term of the form \(h(t)=\sum_jh_j\phi (t-t_j)\chi_{[t_j, \infty)}(t)\) from space-time samples of the solution \(u\) of an initial value problem in a Hilbert space of functions. In the expression of \(h\), the terms \(h_j\) belong to the Hilbert space, while \(\phi\) is a generic real-valued function with exponential decay at \(\infty\). The design of the sampling strategy takes into account noise in measurements and the existence of a background source.Linear complexity and trace representation of balanced quaternary cyclotomic sequences of prime period \(p\)https://zbmath.org/1526.940172024-02-15T19:53:11.284213Z"Yang, Zhiye"https://zbmath.org/authors/?q=ai:yang.zhiye"Xiao, Zibi"https://zbmath.org/authors/?q=ai:xiao.zibi"Zeng, Xiangyong"https://zbmath.org/authors/?q=ai:zeng.xiangyongSummary: Let \(p=ef+1\) be an odd prime, where \(e \equiv 0 \pmod 4\). A family of balanced quaternary sequences is defined by using the classical cyclotomic classes of order \(e\) with respect to \(p\) in this paper. We derive the formulas for their linear complexity and trace representation over \(\mathbb{Z}_4\) by computing the discrete Fourier transform of these sequences. As an application, the linear complexity and trace representation over \(\mathbb{Z}_4\) are given for two types of specific sequences with low autocorrelation derived from the cyclotomic classes of order 4 and 8, respectively. Furthermore, we also determine the exact linear complexity and minimal polynomial for each sequence of the second type over the finite field \(\mathbb{F}_4\).