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Transactions of the American Mathematical Society

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Multiresolution approximations and wavelet orthonormal bases of $ L\sp 2({\bf 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