Expressions for the Coefficients of Optimal Quadrature Formulas with a Parameter

Abstract

Background: Approximate integration of functions with specific smoothness properties requires special mathematical approaches, notably the optimization of quadrature formulas in the sense of Sard. Existing computational methods are primarily designed for classical Banach or Hilbert spaces. However, modern applied problems often involve functions from more complex functional classes that incorporate additional differential characteristics and specialized parameters. Using traditional algorithms for such classes significantly limits computational accuracy, creating a critical need for developing new, adapted computational schemes.

Purpose: This study aims to develop and theoretically substantiate an analytical method for constructing optimal quadrature formulas for a specialized functional space. The primary focus is on minimizing the integration error by finding the exact values of weight coefficients for a given class of integrands.

Method: The proposed approach is based on finding the extremal function of the error functional for the considered quadrature. This problem is solved by constructing a generalized solution to a linear differential equation supplemented with appropriate boundary conditions. The main novelty lies in the analytical calculation of the error functional’s squared norm, which reduces the original problem to a system of linear algebraic equations for the unknown weights. To solve it accurately, a specially constructed discrete analogue of the differential operator is applied, strictly accounting for the internal specificity of the investigated space.

Results: Mathematical analysis yields explicit analytical expressions for the optimal coefficients of the quadrature formulas. The derived exact relations significantly simplify the practical implementation of the algorithm and completely avoid the computational instability that inevitably arises during the numerical inversion of large matrices. The calculated squared norm of the functional confirms the high reliability of the method and guarantees the achievement of the minimum possible integration error for the entire considered class of functions.

Implication: The obtained results significantly expand the existing theory of quadrature formulas. It is proven that under the appropriate limit transition with respect to the space’s weight parameter, the proposed analytical formulas strictly transform into the well-known fundamental results of Sobolev, Shadimetov, and Kohler, previously obtained for classical spaces.