Paley-Wiener Theorem Research Articles

Real Paley-Wiener Theorems for the Multivariable Bessel Transform

Using the harmonic analysis assoicated with the Bessel operator Lα on Ωn = (0,+∞). We establish real Paley-Wiener theorems and we give a necessary and sufficient condition on a function f in order to have a multivariable Bessel transform vanishing on neigberhood of the origin, on symmetric body and a on polynomial domain.

Characterization of the support for the hypergeometric Fourier transform of the W-invariant functions and distributions on R d and Roe’s theorem

In this paper, we establish real Paley-Wiener theorems for the hypergeometric Fourier transform on . More precisely, we characterize the functions of the generalized Schwartz space and of , , whose hypergeometric Fourier transform has bounded, unbounded, convex, and nonconvex support. Finally we study the spectral problem on the generalized tempered distributions . MSC:35L05, 22E30.

Evaluation formulas for generalized Wiener integrals with drift and initial distribution

Let C[0,t] denote a generalized Wiener space, the space of realvalued continuous functions on the interval [0,t]. Let h be of bounded variation with h � 0 a.e. on [0,t] and a be a function on [0,t]. Define a stochastic process Z : C[0,t] × [0,t] → R by Z(x,s )= s 0 h(u)dx(u )+ x(0) + a(s) for x ∈ C[0,t] and s ∈ [0,t]. In this paper we investigate the distribution of Z, and establish a generalized Wiener integration theorem and a generalized Paley-Wiener theorem. As applications of the theorems we evaluate generalized Wiener integrals of various functions on C[0,t].

The classical and approximate sampling theorems and their equivalence for entire functions of exponential type

It is shown that three versions of the sampling theorem of signal analysis are equivalent in the sense that each can be proved as a corollary of one of the others. The theorems in question are the sampling theorem for functions belonging to the Bernstein space Bσ2, the sampling theorem for functions in Bτ∞, 0<τ<σ, and the approximate sampling theorem for non-bandlimited function. One essential difference to an earlier paper of two of the authors is the avoidance of the deep Paley–Wiener theorem of Fourier analysis.

On the range of the generalized Fourier transform associated with a Cherednick type operator on the real line

In this work, we establish the real Paley–Wiener theorem for the generalized Fourier transform on R. Therefore, we study the connection between the real Paley–Wiener theorem and local spectral theory. Finally, we generalize Roe’s theorem.

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

Entire [formula omitted]-functions of exponential type

We characterize the entire functions of exponential type, whose restriction to the real line is in Lp, in different ways: by the usual classical Paley–Wiener growth estimates in the complex plane, by Bernstein inequalities using derivatives or differences, by Lp-growth properties of iterated derivatives or differences, and by support properties of the Fourier image. We also establish Paley–Wiener theorems for the Fourier series of a function on the circle.

Intertwining Operators Associated with a Singular Integro-Differential Operator on the Real Line and Certain of Their Applications

We consider a singular integro-differential operator Λ on the real line which includes as a particular case the Dunkl operator associated with the reflection group Z2 on R. We build intertwining operators of Λ and its dual Λ into the first derivative operator d/dx. Using these intertwining operators, we firstly establish a PaleyWiener theorem for the Fourier transform associated to Λ, and secondly introduce a generalized convolution on R tied to Λ.

Spectral theorems associated with the Jacobi–Cherednik operator

In this paper, we establish real Paley–Wiener theorems for the Opdam–Cherednik transform on R. More precisely, we characterize the functions in the generalized Schwartz space Sk,k′r(R) and in LAk,k′p(R) whose Opdam–Cherednik transform has bounded, unbounded, convex and nonconvex support. Finally we study the spectral problems on the generalized tempered distributions Sk,k′′2(R).

Paley-Wiener and Boas Theorems for the Quaternion Fourier Transform

This paper establishes a real Paley-Wiener theorem to characterize the quaternion-valued functions whose quaternion Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the quaternion Fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator.

Time evolution of excitations in normal Fermi liquids

We inspect the initial and the long time evolution of excitations a Fermi liquids by analyzing the time behavior of the electron spectral function. Focusing on the short-time limit we study the electron-boson model for the homogenous electron gas and apply the first order (in boson propagator) cumulant expansion of the electron Green's function. In addition to a quadratic decay in time upon triggering the excitation, we identify non-analytic terms in the time expansion similar to those found in the Fermi edge singularity phenomenon. We also demonstrate that the exponential decay in time in the long-time limit is inconsistent with the GW approximation for the self-energy. The background for this is the Paley-Wiener theorem of complex analysis. To reconcile with the Fermi liquid behavior an inclusion of higher order diagrams (in the screened Coulomb interaction) is required.

A comparison of Paley–Wiener theorems for real reductive Lie groups

Abstract In this paper we make a detailed comparison between the Paley–Wiener theorems of J. Arthur and P. Delorme for a real reductive Lie group G. We prove that these theorems are equivalent from an a priori point of view. We also give an alternative formulation of the theorems in terms of the Hecke algebra of bi-K-finite distributions supported on K, a maximal compact subgroup of G. Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules.

Paley–Wiener criterion in linear canonical transform domains

The Paley–Wiener criterion involving the issue of the physical realizability of any linear time-invariant (LTI) system / filter is well known in the literature. This criterion gives us the frequency domain condition on the magnitude of the transfer function of the LTI system. In this letter, we extend this criterion to linear canonical transform domains.

Об одном обобщении теоремы Пэли - Винера для весовых пространств Харди

We consider the Hardy space �� p (C+) in the half-plane with an exponential weight. In this space we study the analytic continuation from the boundary. In the previous works for the case p ∈ (1, 2) a result on analytic continuation from the imaginary axis was obtained, and it was a generalization of Paley-Wiener theorem. But for many applications the case p = 1 is more interesting. For this case in the paper we obtain estimates for a function satisfying certain standard conditions.

The Paley-Wiener theorem in the language of Taylor expansion coefficients

The Paley-Wiener theorem in the language of Taylor expansion coefficients

Paley–Wiener theorems and uncertainty principles for the windowed linear canonical transform

In a recent paper, the authors have introduced the windowed linear canonical transform and shown its good properties together with some applications such as Poisson summation formulas, sampling interpolation, and series expansion. In this paper, we prove the Paley–Wiener theorems and the uncertainty principles for the (inverse) windowed linear canonical transform. They are new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Scattering and inverse scattering for a left-definite Sturm–Liouville problem

This work develops a scattering and an inverse scattering theory for the Sturm–Liouville equation −u″+qu=λwu where w may change sign but q⩾0. Thus the left-hand side of the equation gives rise to a positive quadratic form and one is led to a left-definite spectral problem. The crucial ingredient of the approach is a generalized transform built on the Jost solutions of the problem and hence termed the Jost transform and the associated Paley–Wiener theorem linking growth properties of transforms with support properties of functions.One motivation for this investigation comes from the Camassa–Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for −u″+14u=λwu.

On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn

Abstract For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L 2-bounds in a domain of holomorphy in ℂn are obtained with the aid of L. Hörmander’s method of L 2-bounds for the $$\bar \partial$$ operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.

Weighted [formula omitted] Paley–Wiener Theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces

We study the weighted l1 Paley–Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra equation of scalar type in a Hilbert space. The results extend to weighted l1 stability some recent l1 stability theorem due to the author (Numer. Math. 109 (2008) 571–595). Our theoretical results are applied to investigate the asymptotic behavior as tn→∞ of the numerical solutions of certain integro-differential equations in Hilbert space.

Fourier transforms of spherical distributions on compact symmetric spaces

In our previous articles [27] and [28] we studied Fourier series on a symmetric space $M=U/K$ of the compact type. In particular, we proved a Paley-Wiener type theorem for the smooth functions on $M$, which have sufficiently small support and are $K$-invariant, respectively $K$-finite. In this article we extend these results to $K$-invariant distributions on $M$. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. Finally, we characterize the singular support of a distribution in terms of its Fourier transform, and we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator. The extension from symmetric spaces of compact type to all compact symmetric spaces is sketched in an appendix.