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 , 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.