Multiresolution and wavelets

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L 2 (ℝ d ), let S k be the 2 k -dilate of S ( k ∈ℤ). A necessary and sufficient condition is given for the sequence { S k } k∈ℤ to fom a multiresolution of L 2 (ℝ d ). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.