Multiresolution approximations and wavelet orthonormal bases of L2(R)

Abstract

A multiresolution approximation is a sequence of embedded vector spaces (Vj)jΕZ 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 (√2jΨ (2jx - k))(k,j)ΕZ2 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.