ON WAVELETS AND PREWAVELETS WITH VANISHING MOMENTS IN HIGHER DIMENSIONS

Using an approximation theory approach, we prove that a scaling functi on l with suitable polynomial decay satisfies the Strang-Fix condition of order r is an element of N if and only if the elements of any prew avelet set {psi v}(v is an element of E), with polynomial decay of th e same order have vanishing integral moments up to order r-1. An analo gous equivalence is established that does not involve any assumptions concerning decay; this yields a new characterization of the rate of L- 2-approximation of (stationary and nonstationary) multiresolution anal yses in terms of a corresponding prewavelet set. Furthermore, we show that the existence of a scaling Function with polynomial decay implies the existence of both an orthonormal scaling function and a wavelet s et with polynomial decay of the same order. Several known construction s of wavelets and prewavelets are discussed in this respect. (C) 1997 Academic Press.