Wiener-type Theorem Research Articles

Segal–Bargmann Transform and Paley–Wiener Theorems on Motion Groups

We study the Segal–Bargmann transform on a motion group ℝ^n ⋉ K , where K is a compact subgroup of \mathit{SO}(n) . A characterization of the Poisson integrals associated to the Laplacian on ℝ^n ⋉ K is given. We also establish a Paley–Wiener type theorem using complexified representations.

Donoho–Stark and Paley–Wiener theorems for the G-transform

This paper deals with a class of G-transforms g ( x ) = G ( f ) ( x ) : = ∫ 0 ∞ G 2 n , 2 m m , n ( x y | ( a n ) , − ( a ¯ n ) ( b m ) , − ( b ¯ m ) ) f ( y ) d y with the Meijer G-function as the kernel. We prove the Donoho–Stark uncertainty principle for the G-transform. We also give a characterization of the image of functions that have compact support (Paley–Wiener type theorem) under this class of transforms in case n = 0 .

Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces

We prove that Huygens' principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek-Ricci spaces. We also prove a Paley-Wiener type theorem for the inverse of the Helgason Fourier transform on Damek-Ricci spaces.

A Wiener type theorem for (U(p,q) ,Hn)

A Wiener type theorem for (U(p,q) ,Hn)

Laplace hyperfunctions

We introduce the Laplace hyperfunctions generalizing all the Laplace transformable generalized functions. Also, we study various properties of the Laplace transform on Laplace hyperfunctions including the Paley–Wiener type theorem for the Laplace transform.

Real Paley–Wiener theorems for the Koornwinder–Swarttouw q-Hankel transform

We derive two real Paley–Wiener theorems in the setting of quantum calculus. The first uses techniques due to Tuan and Zayed [V.K. Tuan, A.I. Zayed, Paley–Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl. 266 (1) (2002) 200–226] in order to describe the image of the space L q 2 ( 0 , R ) under Koornwinder and Swarttouw q-Hankel transform [T.H. Koornwinder, R.F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1) (1992) 445–461] and contains as a special case a description of the domain of the q-sampling theorem associated with the q-Hankel transform [L.D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (4) (2005) 1197–1203]. The second characterizes the image of compactly supported q-smooth functions under a rescaled version of the q-Hankel transform and is a q-analogue of a recent result due to Andersen [N.B. Andersen, Real Paley–Wiener theorems for the Hankel transform, J. Fourier Anal. Appl. 12 (1) (2006) 17–25].

Paley–Wiener Theorems for the -Spherical Transform: An Overview

Several generalizations of the spherical Fourier transform on Riemannian symmetric spaces have emerged in the last decades. All these integral transforms can be viewed as special instances of the Θ-spherical Fourier transform. One of the main questions related to any integral transform is the description of the image of classical function spaces. In this article we discuss several examples of Paley–Wiener type theorems, leading up to and motivating the Paley–Wiener theorem for the Θ-spherical transform.

Hilbert spaces of analytic functions of infinitely many variables

We study spaces of analytic functions generated by homogeneous polyno- mials from the dual space to the symmetric Hilbertian tensor product of a Hilbert space. In particular, we introduce an analogue of the classical Hardy space H 2 on the Hilbert unit ball and investigate spectral decomposition of unitary operators on this space. Also we prove a Wiener-type theorem for an algebra of analytic functions on the Hilbert unit ball.

Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are proved and commented. It is shownt that the shift asymptotics ( S -asymptotics) of a hyperfunction f is determined by the ordinary asymptotics of (f \star \mathcal K) (x) as x \rightarrow \infty , where \mathcal K is Hörmaner’s kernel. Moreover, Wiener-type theorems are used for the asympthotic analysis of solutions to some (pseudo-)differential equations.

Paley–Wiener-Type Theorems for a Class of Integral Transforms

A characterization of weighted L2(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley–Wiener-type theorems for these spaces. Unlike the classical Paley–Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm–Liouville boundary-value problems on a half line and on the whole line.

On an index integral transform involving the modified bessel function

An index integral transform with the modified Bessel function Iv(z) in the kernel is introduced. This transform is related to a singular Sturm-Liouville problem on a half line. A Plancherel theorem and a Paley–Wiener-type theorem for the transform are also derived

Paley–Wiener Theorems for Hyperbolic Spaces

We prove a topological Paley–Wiener theorem for the Fourier transform defined on the real hyperbolic spaces SOo(p, q)/SOo(p−1, q), for p, q∈2N, without restriction to K-types. We also obtain Paley–Wiener type theorems for Lσ-Schwartz functions (0<σ⩽2) for fixed K-types.

An Integral Transformation and its Applications to Harmonic Analysis on the Space of Solutions of the Heat Equation

We introduce an integral transformation T defined by (Tf)(x,t) = ∫ e^{−ixy−t|y|^2} f(y)dy,\quad f\in L^1(\boldsymbol R^n) in order to do harmonic analysis on the space of C^∞ solutions of the heat equation. First, the Paley–Wiener type theorem for the transformation T will be given for the C^∞ functions and distributions with compact support. Secondly, as an application of the transformation the solutions of heat equation given on the torus T^n will be characterized. Finally, we represent solutions of the heat equation as an infinite series of Hermite temperatures, which are to be defined as the images of Hermite polynomials under the transformation T .

Wiener type theorem for regular operators of bounded support

Wiener type theorem for regular operators of bounded support

A Wiener Type Theorem for Dirichlet Series

A famous theorem of Wiener states the conditions under which the reciprocal of a function with an absolutely convergent Fourier series also has an absolutely convergent Fourier series. We offer an elementary proof of the fact, first proven in [2], that if $F(s)$ has an absolutely convergent Dirichlet series then $1/F(s)$ has an absolutely convergent Dirichlet series if and only if $\left | {F(s)} \right |$ is bounded away from zero in the closed right half-plane.

A Wiener type theorem for Dirichlet series

A famous theorem of Wiener states the conditions under which the reciprocal of a function with an absolutely convergent Fourier series also has an absolutely convergent Fourier series. We offer an elementary proof of the fact, first proven in [2], that if F ( s ) F(s) has an absolutely convergent Dirichlet series then 1 / F ( s ) 1/F(s) has an absolutely convergent Dirichlet series if and only if | F ( s ) | \left | {F(s)} \right | is bounded away from zero in the closed right half-plane.

A new proof of a Paley—Wiener type theorem for the Jacobi transform

which generalizes the Mehler-Fok transform, was studied by Titchmarsh [23, w 17], Olevskii [21], Braaksma and Meulenbeld [2], Flensted--Jensen [9], [11, w and w and Flensted--Jensen and Koornwinder [12]. Some papers by Ch6bli [3], [4], [5] deal with a larger class of integral transforms which includes the Jacobi transform. An even more general class was considered by Braaksma and De Shoo [24]. In the present paper short proofs will be given of a Paley--Wiener type theorem and the inversion formula for the Jacobi transform. The L2-theory, i.e. the Plancherel theorem, is then an easy consequence. These results were earlier obtained by Flensted--Jensen [9], [11, w and by Ch6bli [5]. However, to prove the Paley-Wiener theorem these two authors needed the L2-theory, which can be obtained as a corollary of the Weyl S t o n e T i t c h m a r s h K o d a i r a theorem about the spectral decomposition of a singular Sturm--Liouville operator (cf. for instance Dunford and Schwartz [6, Chap. 13, w The proofs presented here exploit the properties of Jacobi functions as hypergeometric functions and no general theorem needs to be invoked. Furthermore, it turns out that the Paley--Wiener theorem, which was proved by Flensted---Jensen [11, w for real c~, fi, ~ > 1, holds for all complex values of ~ and ft. The key formula in this paper is a generalized Mehler formula