Stability of localized operators
Abbreviated Journal Title
J. Funct. Anal.
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
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 of Functional Analysis
"Stability of localized operators" (2009). Faculty Bibliography 2000s. 2132.