Construction of Approximation Formulas for Analytic Functions by Mathematical Optimization

Abstract

We describe our methods developed recently to construct formulas for approximating functions and numerical integration. Our methods are developed basically for improving the numerical methods with variable transformations known as the double-exponential (DE) Sinc methods for function approximation and DE formulas for numerical integration. They are based on the sinc interpolation and trapezoidal rule on the real axis, respectively. It has been known that they are nearly optimal on Hardy spaces with single- or double-exponential weights, which are regarded as spaces of transformed functions by variable transformations. With a view to an optimal formula in the weighted Hardy space for a general weight, the authors propose a simple method for obtaining sampling points for approximating functions and numerical integration. This method is based on a minimization problem of a discrete energy. By solving the problem with a standard optimization technique, we obtain sampling points that realize accurate formulas for approximating functions and numerical integration. From some numerical examples, we can observe that the constructed formulas outperform the existing formulas.