Composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions

The computation of wavelet coefficients of a function typically requires the computation of a large number of integrals. These integrals represent the inner product of that function with a wavelet function on different scales, or with the corresponding scaling function on a fine scale. We develop quadrature rules for those integrals that converge fast for piecewise smooth and singular functions. They do not require the evaluation of the scaling function, and the convergence does not depend on the smoothness of that function. The analysis and computation is based completely on the filter coefficients that define the scaling function. An application is presented from the field of electromagnetics, involving the inner product of a singular function with two-dimensional tensor-product wavelets.