QUADRATURE-FORMULAS AND ASYMPTOTIC ERROR EXPANSIONS FOR WAVELET APPROXIMATIONS OF SMOOTH FUNCTIONS

This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the constr uction 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 th eir construction using monomials is ill-conditioned, also a modified, well-conditioned construction using Chebyehev polynomials is presented . The second part of the paper deals with pointwise asymptotic error e xpansions of wavelet approximations of smooth functions. They are used to derive asymptotic interpolating properties of the wavelet approxim ation and to construct a convergence acceleration algorithm. This is i llustrated with numerical examples.