Transactions of the American Mathematical Society

Author(s): Qiyu Sun
Journal: Trans. Amer. Math. Soc. 359 (2007), 3099-3123.
MSC (2000): Primary 42C40, 41A65, 41A15
Posted: January 26, 2007
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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 ${\mathcal W}:=\{(a(j-j'))_{j,j'\in\mathbf{Z}^d}: \sum_{j\in \mathbf{Z}^d} \vert a(j)\vert<\infty\}$ . 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 non-commutative algebras to have a similar property to Wiener's lemma for the commutative algebra ${\mathcal W}$ . 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.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42C40, 41A65, 41A15

Qiyu Sun
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: qsun@mail.ucf.edu

DOI: 10.1090/S0002-9947-07-04303-6
PII: S 0002-9947(07)04303-6
Keywords: Wiener's lemma, Banach algebra, inverse of infinite matrices
Received by editor(s): April 15, 2005
Posted: January 26, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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