Potential Theoretic Approach to Design of Accurate Numerical Integration Formulas in Weighted Hardy Spaces

Abstract

We propose a method for designing accurate numerical integration formulas on weighted Hardy spaces, which are regarded as spaces of transformed integrands by some useful variable transformations such as the double-exponential transformation. We begin with formulating an optimality of numerical integration formulas in the space by using the norms of the error operators corresponding to those formulas. Then, we derive an expression of the minimum value of the norms, which gives a criterion for an optimal sequence of sampling points for numerical integration. Based on the expression, we propose an algorithm designing accurate numerical integration formulas on the space by a potential theoretic approach. The effectiveness of the designed formulas is supported by some numerical examples.