An optimal approximation formula for functions with singularities

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 wμ(z)=(1−z2)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, whereas μ is restricted as 0<μ<μ∗ for a certain constant μ∗ in the existing result. We also provide the results of numerical experiments to show the performance of the proposed formula.