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 sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.