An optimal approximation formula for functions with singularities

TL;DR

This paper introduces an optimal approximation formula for analytic functions with endpoint singularities within a Hardy space framework, extending previous results to all positive singularity parameters and demonstrating its effectiveness through numerical experiments.

Contribution

It presents a new optimal approximation formula applicable for any positive singularity parameter, surpassing previous restrictions and validated by numerical results.

Findings

The proposed formula is optimal for all $ > 0$.

Numerical experiments confirm the high performance of the approximation.

The method extends the applicability of approximation formulas to broader singularity cases.

Abstract

We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such functions,we consider a Hardy space with the weight given by $w_{\mu}(z) = (1-z^{2})^{\mu/2}$ for $\mu > 0$, and formulate the optimality of an approximation formula for the functions in the space. Then, we propose an optimal approximation formula for the space for any $\mu > 0$ as opposed to existing results with the restriction $0 < \mu < \mu_{\ast}$ for a certain constant $\mu_{\ast}$. We also provide the results of numerical experiments to show the performance of the proposed formula.