E. K. Narayanan, A. Sitaram
We prove a Wiener-Tauberian theorem for $L^1$-spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz theorem for complex groups. As a corollary we obtain a Wiener-Tauberian type theorem for for compactly supported distributions.
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We prove a genuine analogue of Wiener Tauberian theorem for integrable functions on $\mathrm {SL}(2, \R).$
We prove a genuine analogue of Wiener Tauberian theorem for hypergeometric transforms. As an application we prove analogue of Furstenberg theorem on Harmonic functions.
The theorem is proved that generalizes the Gelfand generalization of the Paley-Wiener tauberian theorem to general abelian topological semigroups with invariant measure. Several corollaries of this theorem are given.
We prove a genuine analogue of Wiener Tauberian theorem for $L^1(G//K)$, where G is a semisimple Lie group of real rank one. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit.
Wiener's Tauberian theorem in classical and quantum harmonic analysis
本文从极限函数视角研究Wiener Tauberian定理，提出多个新版本，并证明了量子调和分析意义下的算子类似定理，进而刻画了严格大于紧算子类的慢振荡算子类，最后探讨了统一版本及其应用。
We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization ope…
We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.
Let $X$ be a symmetric space of the noncompact type. The goal of the paper is to construct in the space $L_{2}(X)$ nearly Parseval frames consisting of functions which simultaneously belong to Paley-Wiener spaces and to Schwartz space on $X$. We call them Paley-Wiener-Schwartz frames in $L_{2}(X)$. These frames are used to characterize a family of Besov spaces on $X$. As a part of our construction we develop on $X$ the so-called average Shannon-type sampling.
We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X = G/K$. We prove a characterisation of their range. In fact, from Delorme's Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener-Schwartz theorems for sections.
In this paper, we establish real Paley-Wiener theorems for the Dunkl transform on Rd. More precisely, we characterize the functions in the Schwartz space S(IRd) and in L2k(Rd) whose Dunkl transform has bounded, unbounded, convex and nonconvex support.
A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions
本文证明了凸函数上光滑赋值的 Paley-Wiener-Schwartz 定理，通过 Fourier-Laplace 变换刻画连续对偶上平移不变赋值，并以此推导平移群表示的光滑向量积分表示，最终实现闭和仿射不变子空间的完整分类及混合 Monge-Ampère 算子赋值的密度结果。
The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the Schwartz class of rapidly decreasing functions. The image of the space of all tempered distributions is also considered and a Paley-Wiener theorem for the windowed Fourier transform is proved.
We first characterize the image of the compactly supported smooth even functions under the q-Weinstein transform as a subspace of the Schwartz space. We then describe the space of smooth $L_{α, q, a}^{2}$-functions whose q-Weinstein transform has compact support as a subspace of the space of $L_{α, q, a}^{2}$-functions.
Real Paley-Wiener theorems in spaces of ultradifferentiable functions
本文在超可微函数类及其相关L^p空间中建立实Paley-Wiener定理，利用Gabor变换、Fourier变换和Wigner变换给出多变量情形下的完整刻画，并讨论了非紧支撑时用多项式处理的相关结果。
We deduce a proof of the isomorphism theorem for certain closed subspace $\mc S^p_Γ(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric space $X$ where $Γ$ is a finite subset of $\what{K}_M$. The Fourier transform considered is the Helgason Fourier transform. Our proof relies only on the Paley-Wiener theorem for the corresponding class of functions and hence it does not use the complicated higher asymptotics of the elementary spherical functions.
In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.
On almost convergence on locally compact abelian groups
本文研究局部紧阿贝尔群上有界可测函数的几乎收敛方法，利用拓扑不变均值定义该概念，给出解析与泛函解析两类充要条件，并证明了整数与实数上的复Tauberian定理。
Non-Wiener groups with a Gelfand pair
本文证明，若非顺从局部紧群G具有紧子群K使得(G,K)为Gelfand对，且G存在拓扑不可约且不可酉化的边界表示，则G不是Wiener群，即其傅里叶变换不满足Wiener Tauberian定理的类比。作为应用，证明了具有无穷多端的连通局部有限图的非紧边界传递自同构群，或非阿基米德局部域上的非交换分裂约化代数群，都不是Wiener群。
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
Real Paley-Wiener theorems and local spectral radius formulas
本文基于反演定理系统建立了 R^d 上 Fourier 变换的实 Paley-Wiener 理论，证明其可精确重建函数支集并有望解释为局部谱半径公式，同时梳理了相关文献。
We elucidate the relation between the two ways of formulating causality in nonlocal quantum field theory: using analytic test functions belonging to the space $S^0$ (which is the Fourier transform of the Schwartz space $\mathcal D$) and using test functions in the Gelfand-Shilov spaces $S^0_α$. We prove that every functional defined on $S^0$ has the same carrier cones as its restrictions to the smaller spaces $S^0_α$. As an application of this result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular generalized functions of tempered growth and obtain the corresponding ex…
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
On the existence problem of regular Gabor frames
本文针对密度大于1的格点Λ，证明了具有连续Zak变换的函数不能生成Gabor框架，从而否定了Schwartz空间等函数类中Gabor框架的存在性。
A new generalized function space in which all Gelfand-Shilov classes $S^{\prime 0}_α$ ($α>1$) of analytic functionals are embedded is introduced. This space of {\it ultrafunctionals} does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on $R^k$ is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the n…
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