Gauss quadrature rules for partial support and logarithmic singular integrals with compactly supported wavelets

Wavelet Gauss quadrature methods are effective approaches to evaluate the inner products with wavelets or with associated scaling functions. By a “lifting trick [A. Barinka, T. Barsch, S. Dahlke, et al., Some remarks on quadrature formulas for refinable functions and wavelets, Z. Angew. Math. Mech. 81 (12) (2001) 839-885]”, the compactly supported wavelets and the associated scaling functions can be used as weighted functions. Gauss quadrature rules for wavelets on partial support are obtained in this paper. Based on the wavelet moments and the partial moments, formulas for calculating the logarithmic wavelet moments are introduced. Finally, Gauss quadrature rules with the product of scaling functions and logarithmic singular functions as the weight function are constructed. The accuracy of the rules is illustrated by two numerical experiments.