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

. A multiresolution approximation is a sequence of embedded vector spaces (Vyjygz for approximating L2(R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 27t-periodic function which is further described. From any multiresolution approximation, we can derive a function >i/(x) called a wavelet such that {V2~Ji//(2Jx - k)),k )6Z2 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.