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.
