Sundaram Thangavelu
We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transform on non-compact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer's formula.
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We investigate the connection between translation bases for Paley-Wiener spaces and exponential Fourier bases for a domain. We apply these results to the characterization of vector-valued time-frequency translates of a Paley-Wiener "window" signal.
We extend the Paley-Wiener pertubation theory to linear operators mapping a subspace of one Banach space into another Banach space.
We give an explicit computation of the Banach envelope for the Paley-Wiener type spaces $E^p, 0<p<1$. This answers a question by Joel Shapiro.
We collect several old and new descriptions of Schatten class Toeplitz operators on the Paley-Wiener space and answer a question on discrete Hilbert transform commutators posed by Richard Rochberg.
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.
Let $M$ be a Riemmanian manifold with bounded geometry. We consider a generalization of Paley-Wiener functions and Lagrangian splines on $M$. An analog of the Paley-Wiener theorem is given. We also show that every Paley-Wiener function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a generalization of the Whittaker-Shannon formula for reconstruction of a Paley-Wiener function from its values on a discrete set. It is shown that every Paley- Wiener function on $M$ is a limit of some linear combinations of fundamental solutions …
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.
It has been shown that Paley-Wiener functions may be recovered from their values on a complete interpolating sequence. This paper explores the same phenomenon, and gives a sufficient condition on a function $ϕ(x)$, called an interpolator, so that scattered translates of this function may be used to interpolate and recover any given Paley-Wiener function.
There is a bounded Hankel operator on the Paley--Wiener space of a disc in $\mathbb{R}^2$ which does not arise from a bounded symbol.
We study the Segal-Bargmann transform on a motion group Rn n K; where K is a compact subgroup of SO(n): A characterization of the Poisson integrals associated to the Laplacian on Rn n K is given. We also establish a Paley-Wiener type theorem using the complexified representations.
We extend Paley-Wiener results in the Bargmann setting deduced earlier by the author together with E. Nabizadeh and C. Pfeuffer to larger class of power series expansions. At the same time we deduce characterisations of all Pilipovi{ć} spaces and their distributions (and not only of low orders as in the earlier work).
An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl-Schrödinger irreducible representations of complexified infinite-dimensional Heisenberg groups.
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.
Sampling on Paley-Wiener spaces on graphs, with particular focus on the infinite-dimensional case
本文针对图上的无穷维Paley-Wiener空间证明了采样定理，实现了稳定框架重构，并指出所有采样集是λ-集的补集，从而为有限几何的Z^n格点和径向树等图提供了稳定采样与重构的充分条件。
In this paper we present generalisations of Paley-Wiener type theorems to Mellin and (Laplace-)Fourier transforms of rapidly decreasing smooth functions with positive support and log-polyhomogeneous asymptotic expansion at zero. This article is based on the thesis \cite{CdC} and uses results borrowed from \cite{RS}, \cite{RS81}, \cite{P79}, \cite{FGD}, \cite{P79}, \cite{SZ}.
In this paper we introduce appropriate associated function to the sequence $M_p=p^{\t p^{\s}}$, $p\in \N$, $\t>0$, $\s>1$, and derive its sharp asymptotic estimates in terms of the Lambert $W$ function. These estimates are used to prove a Paley-Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.
We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform
In this paper we show that functions from the Paley-Wiener amalgam space $(PW,l^1)=\{f\in L^2(\mathbb{R}): \sum\|\hat{f}(ξ+2πm) \|_{L^2([-π,π])} < \infty\}$ enjoy similar recovery properties as the classical Paley-Wiener space. Specifically, if $\{ϕ_α(x): α\in A\}$ is a regular family of interpolators and $\{x_n: n\in \mathbb{Z}\}$ is a complete interpolating sequence for $L^2([-π,π])$, then the family $\{ e^{2πi m x}ϕ_α(x-x_n): m,n\in \mathbb{Z}, α\in A \} $ may be used to recover $f\in(PW,l^1)$.
The $Θ$-spherical functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this article we consider the case of even multiplicity functions. We construct a differential shift operator $D_m$ with smooth coefficients which generates the $Θ$-spherical functions from finite sums of exponential functions. We then use this fact to prove a Paley-Wiener theorem for the $Θ$-spherical transfrom.
The Extended Paley-Wiener Theorem over the Hardy-Sobolev Spaces
本文通过全纯傅里叶变换研究平方可积函数子空间，建立了Hardy-Sobolev空间上的扩展Paley-Wiener定理，并探讨了圆盘与上半平面的联系。
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.
Here we give a new approach to the Paley--Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of "polar-analytic function" in the Mellin setting and Mellin--Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.
A classical result due to Paley and Wiener characterizes the existence of a non-zero function in $L^2(\mathbb{R})$, supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group $M(n)$. We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schrödinger equation on $M(n)$.
Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.
Expansivity and shadowing for composition operators on Paley-Wiener spaces
本文研究了Paley-Wiener空间中复合算子的谱性质及其相关数学特征，探讨了紧性、谱、混沌等关键问题。
We study the Segal-Bargmann transform on the Heisenberg motion groups $\mathbb{H}^n \ltimes K,$ where $\mathbb{H}^n$ is the Heisenberg group and $K$ is a compact subgroup of $U(n)$ such that $(K,\mathbb{H}^n)$ is a Gelfand pair. The Poisson integrals associated to the Laplacian for the Heisenberg motion group are also characterized using Gutzmer's formulae. Explicitly realizing certain unitary irreducible representations of $\mathbb{H}^n \ltimes K,$ we prove the Plancherel theorem. A Paley-Wiener type theorem is proved using complexified representations.
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.
We deduce Paley-Wiener results in the Bargmann setting, which give characterisations of Pilipovi{ć} spaces of low orders, extending the characterisation of a Gr{ö}chenig test function space, deduced earlier by the third author.
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。
Hausdorff Operators on de Branges Spaces and Paley-Wiener spaces
研究了德布拉恩斯空间中豪斯多尔夫算子的有界性和紧性条件，并将其转化为经典积分算子问题，得出了在帕莱-维纳空间中算子的有界性、紧性和核性条件，以及属于希尔伯特-施密特类的条件。
The mixed spectral problem for radial Schrödinger operators and Paley-Wiener spaces
本文研究有限区间径向薛定谔算子的逆谱问题，通过探讨Paley-Wiener空间与势函数的联系，借助完全指数系统，实现由较小区间的势函数和特征值确定整个区间的势函数。
We describe the complete interpolating sequences for the Paley-Wiener spaces $L^p_π$ ($1<p<\infty$) in terms of Muckenhoupt's $(A_p)$ condition. For $p=2$, this description coincides with those given by Pavlov (1979), Nikol'skii (1980), and Minkin (1992) of the unconditional bases of complex exponentials in $L^2(-π,π)$. While the techniques of these authors are linked to the Hilbert space geometry of $L^2_π$, our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted $L^p$ spaces of functions and sequences.
We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary represent…
We prove several Paley--Wiener-type theorems related to the spherical transform on the Gelfand pair $\big(H_n\rtimes U(n),U(n)\big)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in ${\mathbb R}^2$, we prove that spherical transforms of $ U(n)$--invariant functions and distributions with compact support in $H_n$ admit unique entire extensions to ${\mathbb C}^2$, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported fu…
We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data representation, compression, denoising and visualization. These tasks are of great importance to machine learning, complex data analysis and computer vision.
Paley-Wiener Theorem for Probabilistic Frames
本文提出概率框架的帕莱-维纳扰动定理，证明概率测度在特定度量下接近概率框架时仍为概率框架，并获得显式框架界。
The paper studies weak Paley-Wiener properties for group extensions by use of Mackey's theory. The main theorem establishes sufficient conditions on the dual action to ensure that the group has the weak Paley-Wiener property. The theorem applies to yield the weak Paley-Wiener property for large classes of simply connected, connected solvable Lie groups (including exponential Lie groups), but also criteria for non-unimodular groups or motion groups.
A Paley-Wiener theorem for multi-anisotropic ultradifferentiable functions
本研究在多各向同性超可微函数空间中提出实Paley-Wiener定理，并揭示该空间的新性质，丰富了相关函数空间理论。
Besov and Paley-Wiener spaces, Moduli of continuity and Hardy-Steklov operators associated with the group $"ax+b"$
本研究针对直线仿射变换“ax+b”李群的三种表示，刻画了Sobolev、Besov与Paley-Wiener空间的关系，构建了L₂逼近理论并证明了Jackson型不等式。
Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We show that our results give Paley-Wiener theorem for frames for Hilbert spaces.
A version of Paley-Wiener like theorem for connected, simply connected nilpotent Lie groups is proven.
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 a version of the Paley-Wiener theorem of Fourier analysis in the frame of the Mellin transform. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.
Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on noncompact symmetric spaces
本文研究非紧型对称空间上的联合特征空间 Fourier 变换，借助 Poisson 变换与 Helgason Fourier 变换阐明其对偶结构，并给出反演、Plancherel 公式及 Paley-Wiener 定理。
We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley-Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type $a$ whose restriction to the real line belongs to the homogeneous Sobolev space $\dot{W}^{s,p}$ and we call these spaces fractional Paley-Wiener if $p=2$ and fractional Bernstein spaces if $p\in(1,\infty)$, that we denote by $PW^s_a$ and $\mathcal B^{s,p}_a$, respectively. For these spaces we provide a Paley-Wiener type characterization, we remark some facts about the sampling problem in the Hilbe…
Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces
本文研究了Paley-Wiener空间中Hankel算子的Schatten类条件对Nehari定理的影响，证明在特定条件下Nehari定理不成立，并展示了多面体相关的Hardy不等式及其应用。
Real Paley-Wiener theorems for the linear canonical Dunkl transform
本文研究了与线性规范Dunkl变换相关的Sobolev空间及算子性质，建立了实Paley-Wiener定理，刻画了变换支撑于多项式域的平方可积函数，并发展了Boas型Paley-Wiener定理。
The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $Θ$-spherical functions, the correspondint $Θ$-spherical Fourier transform, and the Paley-Wiener theorem for this transform. Up to now such a theorem has only been proven in very special cases, and still, its formulation and proof are very technical. In this paper we do not go into details of the proofs, but present an overview which explains the different examples which have inspired and …
Cyclicity of composition operators on the Paley-Wiener spaces
本文刻画了 Paley-Wiener 空间上复合算子具有循环性的充要条件，并揭示了循环性与复对称性及指数序列完备性之间的联系。
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。
Let $S_1$ and $S_2$ be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets $S_1,S_2$ and exponents $p$ such that the canonical projection $P$ from $PW_{S_1\cup S_2}^p$ to $PW_{S_1}^p$ is a contraction.
It is shown that Paley-Wiener functions on Riemannian manifolds of bounded geometry can be reconstructed in a stable way from some countable sets of their inner products with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. All results are specified in the case of the two-dimensional hyperbolic space in its Poincare upper half-plane realization.
For a sequence of complex numbers $Λ$ we consider the restriction operator $R_Λ$ defined on Paley-Wiener spaces $PW_τ^{p}$ ($1<p<\infty$). Lyubarskii and Seip gave necessary and sufficient conditions on $Λ$ for $R_Λ$ to be an isomorphism between $PW_τ^{p}$ and a certain weighted $l^{p}$ space. The Carleson condition appears to be necessary. We extend their result to $N-$Carleson sequences (finite unions of $N$ disjoint Carleson sequences). More precisely, we give necessary and sufficient conditions for $R_Λ$ to be an isomorphism between $PW_τ^{p}$ and an appropriate sequence space involving di…
Let $μ$ be a measure on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}}\frac{dμ(t)}{1+t^2} < \infty$ and let $a>0$. Assume that the norms $\|f\|_{L^2(\mathbb{R})}$ and $\|f\|_{L^2(μ)}$ are comparable for functions $f$ in the Paley-Wiener space $PW_{a}$ and that $PW_a$ is dense in $L^2(μ)$. We reconstruct the canonical Hamiltonian system $JX' = zHX$ such that $μ$ is the spectral measure for this system.
A Paley type inequality for the Fourier transform on $H^p(H^n);$ the Hardy space on the Heisenberg group, is obtained for $0 < p \leq 1.$
Stability of concentration in the Paley-Wiener space
本文证明了 Paley-Wiener 空间中固定时间-带宽积下集中性的稳定性，通过加权集合差异控制函数距离与区间对称差，并获得了最优指数。
A remark on the Weyl quantisation of Paley-Wiener functions
本文利用 Werner-Young 不等式与量子版 Wiener 分割引理，简洁证明了紧支集 Fourier 变换的缓增分布其 Weyl 量子化属于 Schatten p 类等价于符号 L^p 可积。
Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions
该研究针对高维凸区域上的Paley–Wiener空间，建立Paley–Wiener型Besov空间理论，明确Hankel算子属于Schatten类Sᵖ的充要条件，并拓展了相关结果的适用范围。
Optimal concentration in the Paley-Wiener space
研究帕利-维纳空间中函数在有限测度集\(E\)上的\(L^{2}\)质量集中问题，基于圆上解析三角多项式再生核的普适性极限，分两步证明区间能优化集中，得出区间在该空间中优化集中的结论。
The main result is a Paley's theory for lacunary Fourier series using semigroup-BMO and $H^1$ spaces. This interpretation allows an extension of Paley's theory to general discrete groups, complementing the work of Rudin for abelian groups with a total order, and Lust-Piquard and Pisier's work for lacunary Fourier series with operator-valued coefficients.
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