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Transactions of the American Mathematical Society

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Multiresolution approximations and wavelet orthonormal bases of $L^ 2(\textbf {R})$

Author: Stephane G. Mallat
Journal: Trans. Amer. Math. Soc. 315 (1989), 69-87
MSC: Primary 42C10; Secondary 41A65
MathSciNet review: 1008470
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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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Keywords: Approximation theory, orthonormal bases, wavelets
Article copyright: © Copyright 1989 American Mathematical Society