Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization

TL;DR

This paper introduces a new method for designing approximation formulas for weighted analytic functions in Hardy spaces by minimizing discrete energy, leading to near-optimal sampling points and improved approximation accuracy.

Contribution

The paper presents a novel discrete energy minimization approach to determine near-optimal sampling points for function approximation in weighted Hardy spaces.

Findings

The proposed formulas outperform sinc approximation in numerical tests.

The method provides theoretical error estimates for the approximation.

Sampling points are effectively optimized via discrete energy minimization.

Abstract

We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then, we adopt the minimum worst error of the $n$-point approximation formulas in each space for characterizing the optimal sampling points for the approximation. In order to obtain approximately optimal sampling points, we consider minimization of a discrete energy related to the minimum worst error. Consequently, we obtain an approximation formula and its theoretical error estimate in each space. In addition, from some numerical experiments, we observe that the formula generated by the proposed method outperforms the corresponding formula derived with sinc approximation, which is near optimal in each space.