On a Method of Constructing Quadrature Formulas for Computing Hypersingular Integrals

This paper is devoted to constructing quadrature formulas for evaluating singular and hypersingular integrals. For evaluating integrals with weights (1 - t)(gamma 1)(1 + t)(gamma 2), gamma(1), gamma(2) > -1 defined on [-1, 1] we have constructed quadrature formulas uniformly converging on [-1, 1] to the original integral with weights (1 - t)(gamma 1)(1 + t)(gamma 2), gamma(1), gamma(2) >= -1/2 and converging to the original integral for -1 < t < 1 with weights (1 - t)(gamma 1)(1 + t)(gamma 2), gamma(1), gamma(2) > -1. In the latter case, a sequence of quadrature formulas converges to the integral uniformly on [-1 + delta, 1 - delta], where delta > 0 is arbitrarily small. We propose a method for constructing and estimating the errors of quadrature formulas to evaluate hypersingular integrals by transforming quadrature formulas to evaluate singular integrals. We also propose a method for estimating the errors of quadrature formulas for singular integral evaluation based on approximation theory methods. The results obtained have been extended to hypersingular integrals.