Wavelet Calculus and Finite Diierence Operators

This paper shows that the naturally induced discrete diierentia-tion operators induced from a wavelet-Galerkin nite-dimensional approximation to a standard function space approximates diierentiation with an error of order O(h 2d+2), where d is the degree of the wavelet system. The degree of a wavelet system is deened as one less than the degree of the lowest order non-vanishing moment of the fundamental wavelet. We consider in this paper compactly supported wavelets of the type introduced by Daubechies in 1988. The induced diierentia-tion operators are described in terms of connection coeecients which are intrinsically deened functional invariants of the wavelet system (deened as L 2 inner products of derivatives of wavelet basis functions with the basis functions themselves). These connection coeecients can be explicitly computed without quadrature and they themselves have key moment vanishing properties proved in this paper which are dependent upon the degree of the wavelet system. This is the basis for the proof of the principal results concerning the degree of approximation of the diierenitaion operator by the wavelet-Galerkin discrete diierentiation operator.