WAVELET CALCULUS AND FINITE-DIFFERENCE OPERATORS

This paper shows that the naturally induced discrete differentiation o perators induced from a wavelet-Galerkin finite-dimensional approximat ion to a standard function space approximates differentiation with an error of order O(h2d+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 intro duced 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 L2 in ner 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-vanis hing properties proved in this paper which 'are dependent upon the deg ree of the wavelet system. This is the basis for the proof of the prin cipal results concerning the degree of approximation of the differenti ation operator by the wavelet-Galerkin discrete differentiation operat or.