On an optimal quadrature formula in the sense of Sard (Q634732)

Let \(K_2 = \{\varphi:\,[0,1] \to \mathbb R\); \(\varphi'\) absolutely continuous and \(\varphi'' \in L_2(0,1)\}\) be the Hilbert space with the semi-norm \[ \|\varphi\| = \bigg(\int_0^1 (\varphi''(x) + 1)^2\,dx\bigg)^{1/2}. \] Then the quadrature formula \[ \int_0^1 \varphi(x)\, dx \approx \sum_{\nu =0}^N C_{\nu}\,\varphi(x_{\nu}) \] has the error functional \[ e(x) = \chi_{[0,1]}(x) - \sum_{\nu =0}^N C_{\nu}\,\delta(x-x_{\nu}). \] A minimization of the norm of \(e(x)\) with respect to the coefficients \(C_{\nu}\), when the nodes \(x_{\nu}\in [0,1]\) are fixed, yields an optimal quadrature formula in the sense of \textit{A. Sard} [Am. J. Math. 71, 80--91 (1949; Zbl 0039.34104)]. In this paper, the authors construct the optimal quadrature formula in the sense of Sard in the Hilbert space \(K_2\). For equidistant nodes \(x_{\nu} = \nu/N\), the corresponding optimal coefficients \(C_{\nu}\) are explicitly given. Further, the order of convergence of the optimal quadrature formula is investigated. The asymptotic optimality of such a formula, which is exact for \(\sin x\) and \(\cos x\), is shown, too. Some numerical examples are given.