Multiresolution approximations and wavelet orthonormal bases of L^2(R)

TLDR: In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.

Abstract: A multiresolution approximation is a sequence of embedded vector spaces   V j  jmember Z for approximating L 2 (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   √  2 j ψ(2 j x −k)   (k ,j)member Z 2 is an orthonormal basis of L 2 (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 H s .