Stability of Localized Operators

By Chang Eon Shin, Qi-yu Sun. 2008

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Let lp, 1⩽p⩽∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the lp-stability of the convolution operator Ca for different 1⩽p⩽∞ are equivalent to each other, i.e., if Ca has lp-stability for some 1⩽p⩽∞ then Ca has lq-stability for all 1⩽q⩽∞. 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 lp-stability (or Lp-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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/09bbc847fa6855bbef36ee33d5ac190876d1112d

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Deformation of Gabor systems (2013, background): “Several stability results are built on his techniques [37, 1, 35, 39, 38].”
Wiener’s Lemma for Infinite Matrices II (2007, background): “[45] Q. Sun, Frames in spaces with finite rate of innovations, Adv.”