Title

Wiener'S Lemma For Infinite Matrices

Keywords

Banach algebra; Inverse of infinite matrices; Wiener's lemma

Abstract

The classicalWiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W := {(a(j -j'))j,j'εZd : σjεZd |a(j)| < ∞}. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those noncommutative algebras to have a similar property to Wiener's lemma for the commutative algebraW. In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener's lemmas for those matrix algebras. © 2007 American Mathematical Society.

Publication Date

7-1-2007

Publication Title

Transactions of the American Mathematical Society

Volume

359

Issue

7

Number of Pages

3099-3123

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/S0002-9947-07-04303-6

Socpus ID

48349088218 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/48349088218

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