Completeness of Sets of Shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian conditions

TL;DR

This paper generalizes the completeness of shifted functions in invariant Banach spaces of tempered distributions using Tauberian conditions, enabling approximation of identity operators with finite rank within these spaces.

Contribution

It extends previous results by employing Tauberian conditions to show density of translates in Banach spaces and constructs finite rank operators for approximation.

Findings

Linear span of translates is dense in certain Banach spaces.

Finite rank operators can approximate the identity within these spaces.

The approach simplifies technical arguments using Fourier analysis methods.

Abstract

The main result of this paper is a far reaching generalization of the completeness result given by V.~Katsnelson in a recent paper [35]. Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper [27] by the authors, combined with the use of Tauberian conditions (i.e. the non-vanishing of the Fourier transform) allow us to show that the linear span of the translates of a single function $g \in {{{\boldsymbol{\mathcal {S}}}({\mathbb{R}}^d)}}$ is a dense subspace of any Banach space satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset $M$ in such a Banach space $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbol B})$ one can construct a finite rank operator, whose range is contained in the linear span of finitely many translates of $g$, and which approximates the identity operator over $M$ up to a given level of precision. The setting of tempered distributions allows to reduce the technical arguments to methods which are widely used in Fourier Analysis. The extension to non-quasi-analytic weights respectively locally compact Abelian groups is left to a forthcoming paper, which will be technically much more involved and uses different ingredients.