Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces

TL;DR

This paper introduces a potential theoretic method to design nearly optimal interpolation formulas for function approximation in weighted Hardy spaces, outperforming existing sinc-based formulas especially with double exponential weights.

Contribution

The paper develops a novel potential theoretic approach to construct almost optimal formulas in weighted Hardy spaces for general weights, surpassing traditional sinc formulas.

Findings

The method produces formulas that outperform DE-Sinc in numerical tests.

Numerical solutions to the optimization problem confirm the near optimality of the formulas.

The approach is applicable to a wide class of weight functions in Hardy spaces.

Abstract

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.