Stability of Localized Operators

TL;DR

This paper investigates the stability of various localized operators, including convolution, matrix, synthesis, and integral operators, demonstrating their stability properties are equivalent across different function spaces and that their inverses are well localized.

Contribution

It establishes the equivalence of $ ext{l}^p$-stability among different classes of localized operators and shows their inverses are well localized, extending stability results beyond convolution operators.

Findings

$ ext{l}^p$-stability of localized operators are equivalent across classes

Inverses of these operators are well localized

Stability results apply to operators in Sjöstrand class, Wiener amalgam space, and with regular kernels

Abstract

Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sj\"ostrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the $\ell^p$- stability (or $L^p$-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.