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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Wiener’s lemma for infinite matrices
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by Qiyu Sun PDF
Trans. Amer. Math. Soc. 359 (2007), 3099-3123 Request permission

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 ${\mathcal W}:=\{(a(j-j’))_{j,j’\in \mathbf {Z}^d}: \ \sum _{j\in \mathbf {Z}^d} |a(j)|<\infty \}$. 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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Additional Information
  • Qiyu Sun
  • Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
  • Email: qsun@mail.ucf.edu
  • Received by editor(s): April 15, 2005
  • Published electronically: January 26, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3099-3123
  • MSC (2000): Primary 42C40, 41A65, 41A15
  • DOI: https://doi.org/10.1090/S0002-9947-07-04303-6
  • MathSciNet review: 2299448