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Journal Article

# Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R)

Stephane G. Mallat
Transactions of the American Mathematical Society
Vol. 315, No. 1 (Sep., 1989), pp. 69-87
DOI: 10.2307/2001373
Stable URL: http://www.jstor.org/stable/2001373
Page Count: 19
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## Abstract

A multiresolution approximation is a sequence of embedded vector spaces (Vj)j∈ Z for approximating L2(R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π-periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that $(\sqrt{2^j}\pi(2^jx - k))_{(k,j)\in Z^2}$ is an orthonormal basis of L2(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 Hs.

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