An optimal approximation formula for functions with singularities

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 $w_μ(z) = (1-z^{2})^{μ/2}$ for $μ> 0$, 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 $μ> 0$ as opposed to existing results with the restriction $0 < μ< μ_{\ast}$ for a certain constant $μ_{\ast}$. We also provide the results of numerical experiments to show the performance of the proposed formula.