On one method of constructing quadrature formulas for computing hypersingular integrals

This paper is devoted to constructing quadrature formulas for singular and hypersingular integrals evaluation. For evaluating the integrals with the 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 the weights $(1-t)^{\gamma_1}(1 + t)^{\gamma_2}$, $\gamma_1$, $\gamma_2\geqslant-1/2$, and converging to the original integral for $-1 < t < 1$ with the 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 evaluating integral uniformly on $[-1 + \delta, 1 -\delta]$, where $\delta>0$ is arbitrarily small. We propose a method for construction and error estimate of quadrature formulas for evaluating hypersingular integrals based on transformation of quadrature formulas for evaluation of singular integrals. We also propose a method of the error estimate for quadrature formulas for singular integrals evaluation based on the approximation theory methods. The results obtained were extended to hypersigular integrals.