Wiener's lemma for infinite matrices

    Authors

    Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    Trans. Am. Math. Soc.

    Keywords

    Wiener's lemma; Banach algebra; inverse of infinite matrices; HOMOGENEOUS TYPE; FRAMES; SPACES; LOCALIZATION; SYMMETRY; ALGEBRAS; Mathematics

    Abstract

    The classical Wiener 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'is an element of Zd) : Sigma(j is an element of Zd) |a(j)| < infinity}. In the study of spline approximation, (diffusion) wavelets and a. ne 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 non-commutative algebras to have a similar property to Wiener's lemma for the commutative algebra W. In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjostrand class, and establish Wiener's lemmas for those matrix algebras.

    Journal Title

    Transactions of the American Mathematical Society

    Volume

    359

    Issue/Number

    7

    Publication Date

    1-1-2007

    Document Type

    Article

    Language

    English

    First Page

    3099

    Last Page

    3123

    WOS Identifier

    WOS:000245149100006

    ISSN

    0002-9947

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