Optimal Quadrature Formulas for Highly Oscillatory Cauchy-Type Integrals

Background: Singular integrals with oscillatory kernels arise in various areas of computational mathematics and applied science. Their numerical evaluation is challenging because singular behavior and rapid oscillations occur simultaneously, which may significantly reduce the accuracy and efficiency of conventional quadrature methods. Existing approaches often treat these features separately and do not fully address the construction of optimal quadrature formulas within a functional-analytic framework. Purpose: The purpose of this study is to construct optimal quadrature formulas for the numerical evaluation of singular integrals with oscillatory kernels and to establish their optimality in an appropriate Sobolev space. Particular emphasis is placed on obtaining explicit quadrature coefficients that are suitable for practical computation. Method: The quadrature problem is formulated as an extremal problem for the corresponding error functional in a Sobolev space. Following the Sobolev method, the quadrature coefficients are determined by minimizing the norm of the error functional over a prescribed class of quadrature formulas. The analytical properties of both the singular integrand and the oscillatory kernel are incorporated directly into the construction. This formulation provides a unified treatment of singularities and rapid oscillations and leads to explicit analytical expressions for the quadrature coefficients. Results: The constructed formulas possess the minimum-norm property of the associated error functional in the underlying Sobolev space. The explicit coefficient representations permit direct numerical implementation and provide accurate evaluation of singular oscillatory integrals without excessive mesh refinement. The numerical performance demonstrates that the proposed approach remains effective in the presence of both singular behavior and rapid oscillations. Implication: The proposed methodology provides a rigorous and computationally efficient framework for constructing optimal quadrature formulas for singular oscillatory integrals. It extends the functional-analytic approach to numerical integration and offers a useful basis for the treatment of related singular integral operators.