Shift-invariant subspaces of Sobolev type

TL;DR

This paper reviews the structure and properties of shift-invariant subspaces within Sobolev spaces, summarizing key results and extending existing theorems to Sobolev-type spaces without proofs.

Contribution

It provides a comprehensive summary of shift-invariant Sobolev spaces, characterizes Bessel sequences, frames, and Riesz families, and extends Fourier multiplier theorems to these spaces.

Findings

Characterization of shift-invariant Sobolev spaces

Extension of Fourier multiplier theorems to Sobolev spaces

Summary of Bessel sequences, frames, and Riesz families in Sobolev contexts

Abstract

This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\in\mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s(\mathbb{R}^n)$, are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier $\left(1-\frac{\Delta}{4\pi^2}\right)^{s/2}f=\mathcal{F}^{-1}\big((1+|t|^2)^{s/2}\widehat{f}(t)\big)$, we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type.