Wavelet Approximation of Deterministic and Random Signals: Convergence Properties and Rates

Multiresolution signal decomposition and wavelet orthonormal bases of' L2(-", -) have received increasing attention in recent years in the mathematical and in the signal and image processing literatures, see [l]. A multiresolution decomposition of L2(, -) is an increasing sequence {V(}& of closed subspaces of L2(--, =) with dense union, empty intersection, and certain translation and scaling properties [l]. The approximation of a fu_nction f~ L2(-, =) at resolution 2i is the orthogonal projection f, offon V I which is computed by using a wavelet orthonormal basis for V I , (I$( ,k( t ) = 2 " 2 ~ ( 2 ' t k ) } ; ~ , generated by a scale function # E L + = , ) by means of dilations and translations. The simplest example is the Haar basis where 4(r)= 1[0, , ] ( t ) has compact support and is discontinuous. There are scale functions which are k-times continuously differentiable with compact