# The extremal function of the optimal quadrature formula with derivative

## The extremal function of the optimal quadrature formula with derivative

## Abstract

Abstract. This work is devoted to the process of constructing an optimal quadrature formula with

derivative in the sense of Sard in the Hilbert space of periodic and complex-valued functions

for numerical calculation of Fourier integrals. Here a quadrature sum consists of a linear combination of

the given function value on a uniform mesh. The error of a quadrature formula is estimated from above by

the functional norm of the error based on the Cauchy-Schwarz inequality. To calculate the norm, the

concept of an extremal function is used. The extremal function corresponding to the error functional is

found using the Riesz representation theorem.

## References

Sobolev S.L., Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974, 808 p.

Haytov A.R. and Khayriev U.N., Optimal quadrature formulas for approximate calculation of integrals with exponential weight, Bulletin of the Institute of Mathematics, Vol. 5, №6, pp.14-22.

Khayriev U.N. Construction of the Exponentially Weighted Optimal Quadrature Formula in a Hilbert Space of Periodic Functions. // Problems of Computational and Applied Mathematics. -Tashkent. 2022, vol. 44, no. 5/1, pp. 134–142.

## Published

## How to Cite

*MODERN PROBLEMS AND PROSPECTS OF APPLIED MATHEMATICS*,

*1*(01). Retrieved from https://ojs.qarshidu.uz/index.php/mp/article/view/517