Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method

TL;DR

This paper develops optimal quadrature formulas exact for exponential-trigonometric functions in Sobolev spaces using Sobolev's method, including error analysis, system solutions, and discrete operator construction.

Contribution

It introduces a novel approach to constructing optimal quadrature formulas for exponential-trigonometric functions using Sobolev's method and discrete analogues of differential operators.

Findings

Explicit formulas for optimal quadrature in specific cases m=1 and m=3.

Construction of a discrete analogue of a differential operator.

Proof of existence and uniqueness of the quadrature formulas.

Abstract

The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space $W_2^{(m,0)}$ is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{d x^{2m}}-1$ is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space $W_2^{(m,0)}$, which based on the discrete analogue $D_m(h\beta)$, is described. Finally, for $m=1$ and $m=3$ the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.