Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions

Abstract

This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions. Several types of quadratures are discussed and compared for different classes of wavelets. Since their construction using monomials is ill-conditioned, also a modified, well-conditioned construction using Chebyshev polynomials is presented. The second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions. They are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm. This is illustrated with numerical examples.