Optimal Quadrature Formulas for Approximate Calculation of Fast Oscillating Integral
DOI:
https://doi.org/10.71310/pcam.4_68.2025.08Keywords:
Sobolev space, optimal coefficients, error functional, extremal functionAbstract
The article investigates optimal quadrature formulas designed for the approximate evaluation of highly oscillatory integrals, which frequently arise in many applied problems of mathematical physics, signal theory, and computational mathematics. The work is based on the formulation and solution of Sard’s problem in a Sobolev space, where quadrature formulas are constructed taking into account not only the values of the integrand but also its derivatives at the nodal points. This approach makes it possible to significantly improve the accuracy of approximation. To determine the optimal coefficients of the quadrature formula, the Sobolev method is applied, which allows one to derive an analytical expression for the norm of the error functional. Based on the use of the extremal function and the Riesz representation theorem, a boundary value problem is constructed, the solution of which yields the explicit form of the optimal coefficients and an exact estimate of the error. The presented results have a solid theoretical foundation and confirm the effectiveness of the proposed method. The obtained optimal quadrature formulas provide high accuracy in computing integrals with highly oscillatory kernels, opening up prospects for their application in numerical methods for solving differential equations, modeling physical processes, and other areas of computational mathematics.
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