Optimal Quadrature Formulas with Trigonometric Weights in a Sobolev Space

Background: The accurate and efficient numerical evaluation of highly oscillatory integrals remains a challenging problem in computational mathematics and applications such as mathematical physics, wave dynamics, optics, and signal processing. Classical approaches, including Filon-, Levin-, and asymptotic-type methods, are effective for highly oscillatory integrals but do not directly address the construction of optimal quadrature formulas in Sobolev spaces or the minimization of the associated error functional. Purpose: This study aims to construct an optimal quadrature formula for highly oscillatory integrals with trigonometric weights in Sobolev spaces, determine its quadrature coefficients, and investigate the behavior of the corresponding error. Method: The integral is decomposed into sine and cosine components and transformed through piecewise integration to obtain a formulation suitable for the Sobolevspace framework. The quadrature coefficients are derived by minimizing the norm of the associated error functional. A system of conditions characterizing the optimal coefficients is established, and explicit expressions are obtained for uniformly distributed nodes. This approach incorporates both the oscillatory structure of the integrand and the functional properties of the underlying Sobolev space into the quadrature construction. Results: Explicit analytical expressions for the optimal quadrature coefficients are obtained. The derived formulas satisfy the required exactness conditions for constant and linear functions. Numerical experiments demonstrate that the quadrature error exhibits a non-monotonic dependence on the oscillation parameter, with substantial error reduction observed for specific parameter values. Implication: The proposed quadrature formula provides a Sobolev-space-based optimal approach for highly oscillatory integrals and extends existing quadrature techniques by incorporating oscillatory behavior directly into the optimization of the approximation error.