Stability of localized operators

    Authors

    C. E. Shin;Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    Wiener's lemma; Stability; Infinite matrix with off-diagonal decay; Synthesis operator; Localized integral operator; Banach algebra; Gabor; system; Sampling; Schur class; Sjostrand class; Kurbatov class; TIME-FREQUENCY ANALYSIS; SHIFT-INVARIANT SPACES; INFINITE MATRICES; INTEGRAL-OPERATORS; WIENERS LEMMA; PSEUDODIFFERENTIAL-OPERATORS; CONTINUITY PROPERTIES; BANACH FRAMES; FINITE RATE; ALGEBRAS; Mathematics

    Abstract

    Let l(P), 1 < = p < = infinity, be the space of all p-summable sequences and C,, be the convolution operator associated with a summable sequence a. It is known that the l(P)-stability of the convolution operator C, for different 1 < = p < = infinity are equivalent to each other, i.e., if C-a has l(p)-stability for some 1 < = p < = infinity then C-a has l(q)-stability for all 1 < = q < = infinity. 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 Sjostrand 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 l(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. (C) 2008 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Functional Analysis

    Volume

    256

    Issue/Number

    8

    Publication Date

    1-1-2009

    Document Type

    Article

    Language

    English

    First Page

    2417

    Last Page

    2439

    WOS Identifier

    WOS:000264684300002

    ISSN

    0022-1236

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