Wavelet calculus and finite difference operators

Abstract:

This paper shows that the naturally induced discrete differentiation operators induced from a wavelet-Galerkin finite-dimensional approximation to a standard function space approximates differentiation with an error of order ${\text {O}}({h^{2d + 2}})$, where d is the degree of the wavelet system. The degree of a wavelet system is defined as one less than the degree of the lowest-order nonvanishing moment of the fundamental wavelet. We consider in this paper compactly supported wavelets of the type introduced by Daubechies in 1988. The induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system (defined as ${L^2}$ inner products of derivatives of wavelet basis functions with the basis functions themselves). These connection coefficients 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 differentiation operator by the wavelet-Galerkin discrete differentiation operator.