Wiener's Tauberian Theorem Research Articles

Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

<abstract><p>For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.</p></abstract>

WIENER TAUBERIAN THEOREMS FOR CERTAIN BANACH ALGEBRAS ON REAL RANK ONE SEMISIMPLE LIE GROUPS

AbstractWe prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G.

Polynomial approximations in a generalized Nyman–Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

A Distributional Proof of $$p$$-Adic Wiener Tauberian Theorem and Approximation by Translates of a Function

A Distributional Proof of $$p$$-Adic Wiener Tauberian Theorem and Approximation by Translates of a Function

A Wiener Tauberian theorem for operators and functions

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 operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)

Abstract In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for L p , 1 ⁢ ( G ) {L^{p,1}(G)} ( 1 ≤ p < 2 {1\leq p<2} ), with G = SL ⁢ ( 2 , ℝ ) {G=\mathrm{SL}(2,\mathbb{R})} .

Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms

AbstractWe prove a genuine analogue of the Wiener Tauberian theorem for , where G is a real rank one noncompact, connected, semisimple Lie group with finite centre. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm, and Y. Weit. We extend this result for hypergeometric transforms and as an application we prove an analogue of Furstenberg theorem on harmonic functions for hypergeometric transforms.

On the boundary behavior of positive solutions of elliptic differential equations

Let $u$ be a positive harmonic function in the unit ball $B_1 \subset \mathbb{R}^n$ and let $\mu$ be the boundary measure of $u$. Consider a point $x\in \partial B_1$ and let $n(x)$ denote the unit normal vector at $x$. Let $\alpha$ be a number in $(-1,n-1]$ and $A \in [0,+\infty) $. We prove that $u(x+n(x)t)t^{\alpha} \to A$ as $t \to +0$ if and only if $\frac{\mu({B_r(x)})}{r^{n-1}} r^{\alpha} \to C_\alpha A$ as $r\to+0$, where ${C_\alpha= \frac{\pi^{n/2}}{\Gamma(\frac{n-\alpha+1}{2})\Gamma(\frac{\alpha+1}{2})}}$. For $\alpha=0$ it follows from the theorems by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For $\alpha=n-1$ it concerns about the point mass of $\mu$ at $x$ and it follows from the Beurling minimal principle. For the general case of $\alpha \in (-1,n-1)$ we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin's approach. Unfortunately this approach works for a ball or a half-space only but not for a general kind of domain. In dimension $2$ one can use conformal mappings and generalise the statement above to sufficiently smooth domains, in dimension $n\geq 3$ we showed that this generalisation is possible for $\alpha\in [0,n-1]$ due to harmonic measure estimates. The last method leads to an extension of the theorems by Loomis, Ramey and Ullrich on non-tangential limits of harmonic functions to positive solutions of elliptic differential equations with Holder continuous coefficients.

Wiener's tauberian theorems for the Fourier–Bessel transformation and uniform approximation by RBF networks of Delsarte translates

We prove analogues of the Wiener and Wiener–Pitt tauberian theorems for the Delsarte translation and the Fourier–Bessel transformation. Through similar functional-analytic techniques, we give conditions on the kernel function (or activation function) for the family of radial basis function (RBF) neural networks obtained upon replacing the usual translation by the Delsarte one, with not necessarily the same smoothing factor in all kernel nodes, to have the universal approximation property in a certain space of continuous functions endowed with the uniform norm.

A Wiener-Tauberian Type Theorem for Arbitrary Normed Algebras with an Approximate Identity

For an arbitrary normed algebra with an approximate identity, we introduce some notions of invertibility of a net in this algebra with respect to the given approximate identity. By means of that, we show a Wiener-Tauberian type theorem in that general situation. An application yields an abstract Wiener-Tauberian theorem for a class of commutative Banach algebras, which implies the classical Wiener-Tauberian theorem for locally compact Abelian groups.

$$L^p$$ L p -Integrability, Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in $L^p(\R^n)$ with $1\leq p\leq 2n/\alpha$ if its Fourier transform is supported by a set of finite packing $\alpha$-measure where $0<\alpha<n$. It is shown that the assertion fails for $p>2n/\alpha$. The result is applied to prove $L^p$ Wiener-Tauberian theorems for $\R^n$ and M(2).

Spectral analysis on $${{\rm SL}(2, \mathbb{R})}$$

Let G be the group $${{\rm SL}(2, \mathbb{R})}$$ . For this group we prove a version of Schwartz’s theorem on spectral analysis for the group G. We find the sharp range of Lebesgue spaces L p (G) for which a smooth function is not mean periodic unless it is a cusp form. Failure of the Schwartz-like theorem is also proved when C ∞(G) is replaced by L p (G) with suitable p. We show that the last result is linked with the failure of the Wiener-tauberian theorem for G.

Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces

We prove an analogue of Schwartz’s theorem on spectral analysis for radial sections of some homogeneous vector bundles on noncompact Riemannian symmetric spaces. We include some results and observations regarding mean periodic functions in this case. We also observe failure of spectral analysis for various Lorentz spaces and Lebesgue spaces of radial sections and relate it with the failure of the Wiener-Tauberian theorems in this setup.

Analogues of the Wiener Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener Tauberian theorem for the L 1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz-type theorem for complex groups. As a corollary we obtain a Wiener Tauberian type result for compactly supported distributions.

Constructive theory of Banach algebras

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra.

Applications of P-adic generalized functions and approximations by a system of p-adic translations of a function

Under some conditions we prove that the convergence of a sequence of functions in the space of P-adic generalized functions is equivalent to its convergence in the space of locally integrable functions. Some analogs are established of the Wiener tauberian theorem and the Wiener theorem on denseness of translations for P-adic convolutions and translations.

A Wiener–Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener–Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337–363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.

Sir Harry Raymond Pitt. 3 June 1914 — 8 October 2005

Sir Harry Pitt worked (as H. R. Pitt) with Norbert Wiener in 1938 on Wiener's general Tauberian theory. Mathematically, he is best known for Pitt's form of Wiener's Tauberian theorem, and as the author of the first (1958) monograph on Tauberian theory. He is otherwise best known for having been Vice–Chancellor of Reading University from 1964 to 1978.

Pompeiu-type theorem associated with the Dunkl operator on the real line

In this work, by using the Wiener-Tauberian Theorem, we establish a Pompeiu-type theorem (which consists to study the uniqueness of some convolution equations solutions) associated with a differential-difference operator 𝒟α, called the Dunkl operator on the real line. Next, we give some applications.

Ergodicity and stability of orbits of unbounded semigroup representations

AbstractWe develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.