Optimal Quadrature Formulas and Interpolation Splines Minimizing the Semi-Norm in the Hilbert Space $$K_{2}(P_{2})$$

Abstract

In this paper we construct the optimal quadrature formulas in the sense of Sard, as well as interpolation splines minimizing the semi-norm in the space \(K_{2}(P_{2})\), where \(K_{2}(P_{2})\) is a space of functions \(\varphi\) which \(\varphi ^{\prime}\) is absolutely continuous and \(\varphi ^{\prime\prime}\) belongs to L 2(0, 1) and \(\int _{0}^{1}{(\varphi ^{\prime\prime}(x) {+\omega }^{2}\varphi (x))}^{2}dx < \infty \). Optimal quadrature formulas and corresponding interpolation splines of such type are obtained by using S.L. Sobolev method. Furthermore, order of convergence of such optimal quadrature formulas is investigated, and their asymptotic optimality in the Sobolev space \(L_{2}^{(2)}(0,1)\) is proved. These quadrature formulas and interpolation splines are exact for the trigonometric functions sinω x and cosω x. Finally, a few numerical examples are included.