Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization

TL;DR

This paper analyzes the near-optimality of specific approximation formulas for analytic functions in weighted Hardy spaces, using duality theory to establish error bounds that match heuristic predictions.

Contribution

It provides a convergence analysis and error bounds for approximation formulas in weighted Hardy spaces, demonstrating their near-optimality via duality for potential energy minimization.

Findings

Approximation formulas are nearly optimal under worst error criteria.

Upper bounds of approximation errors match heuristic asymptotic bounds.

Duality theorem effectively analyzes potential energy minimization problems.

Abstract

We investigate the approximation formulas that were proposed by Tanaka & Sugihara (2019), in weighted Hardy spaces, which are analytic function spaces with certain asymptotic decay. Under the criterion of minimum worst error of $n$-point approximation formulas, we demonstrate that the formulas are nearly optimal. We also obtain the upper bounds of the approximation errors that coincide with the existing heuristic bounds in asymptotic order by duality theorem for the minimization problem of potential energy.