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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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Multiresolution approximations and wavelet orthonormal bases of $L^ 2(\textbf {R})$
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by Stephane G. Mallat PDF
Trans. Amer. Math. Soc. 315 (1989), 69-87 Request permission

Abstract:

A multiresolution approximation is a sequence of embedded vector spaces ${({{\mathbf {V}}_j})_{j \in {\text {z}}}}$ for approximating ${{\mathbf {L}}^2}({\mathbf {R}})$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $2\pi$-periodic function which is further described. From any multiresolution approximation, we can derive a function $\psi (x)$ called a wavelet such that ${(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text {z}}^2}}}$ is an orthonormal basis of ${{\mathbf {L}}^2}({\mathbf {R}})$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space ${{\mathbf {H}}^s}$.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 315 (1989), 69-87
  • MSC: Primary 42C10; Secondary 41A65
  • DOI: https://doi.org/10.1090/S0002-9947-1989-1008470-5
  • MathSciNet review: 1008470