ABSTRACT

The problem of quickly computing integrals (derivatives) of functions arises in various fields. For example, one might want to generate linear systems derived from Petrov-Galerkin schemes for finding the approximate solution of boundary value problems in Numerical Analysis. In this case, one usually has to compute integrals involving partial and normal derivatives induced by the underlying partial differential operator, and L2 inner products for the right hand side, see e.g. [3,4] and the references therein. The construction of wavelet-type functions often requires the computation of L2 inner products as well. Or, since (derivatives) integrals of box splines are again box splines, one can use the calculation of the integrals as a method for evaluating box splines rather than using a recurrence formula. Since the main ingredient of the method proposed here is a refinement equation, all kinds of integrals of products of (derivatives of) wavelet-type functions which are needed for vari­ ous applications can simply be reduced to the terms treated in the sequel. The method is based on function evaluations of a refinable multivariate function at integers, and can therefore also be used for the fast evaluation of (derivatives of) multivariate functions, see [7] for examples including Daubechies’ refin­ able functions and orthonormal wavelets. The need for evaluating functions typically arises in Computer Aided Geometric Design.