On an optimal quadrature formula in the sense of Sard

Abstract

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K 2 (P 2 ). Using S.L. Sobolev's method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (2) 2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x and cos x. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

Optimal quadrature formula in the sense of Sard in K2(P3) space

“…Finally, in Section 6 some numerical results are presented. It should be noted that the results of this paper is a continuation of the results of [12].…”

“…where C ν and x ν (∈ [0, 1]) are coefficients and nodes of formula (1.1), respectively, χ [ Equality (13) is semi-norm and ϕ = 0 if and only if ϕ(x) = c 1 sin x + c 2 cos x + c 3 . The case without the constant term in ϕ(x) was considered in our previous work [12].…”

Optimal quadratures in the sense of Sard in a Hilbert space

“…It should be noted that the results of this paper is a continuation of our previous results [13,14], where these problems have been solved only for some particular values of m ð¼ 2; 3Þ and x ¼ 1.…”

An optimal quadrature formula with derivative in the space $W_2^{(2,1)}$

“…[20,21,22]). In different spaces based on these methods, the Sard problem was studied by many authors, see, for example, [2,3,4,5,6,7,8,11,13,15,20,21,22,23] and references therein.…”

“…From Theorem 1, in the case m = 2, we get the extremal function ψ ℓ for the error functional (5) and it has the form…”