Optimal Quadrature Formula in the Sense of Sard in K2(p3) Space
نویسندگان
چکیده
We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using 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 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sinx, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.
منابع مشابه
Optimal Quadrature Formulas for the Cauchy Type Singular Integral in the Sobolev Space
Abstract This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in (2) 2 ( 1,1) L − S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete analogue of the operator 4 4 / d dx we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the optimal coefficients are obtained. Finally, in...
متن کاملOptimal Quadrature Formulas with Derivative in the Space
Abstract This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space ( ) 2 (0,1) m L . In this paper the quadrature sum consists of values of the integrand and its first derivative at nodes. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number 1 N m ≥ ...
متن کاملOPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W ( m , m − 1 ) 2 SPACE
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W (m,m−1) 2 [0, 1] space for calculating Fourier coefficients. Using S. L. Sobolev’s method we obtain new optimal quadrature formulas of such type for N + 1 ≥ m, where N + 1 is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the...
متن کاملOn the Universalityof the Gaussian Quadrature FormulaKnut
Several deenitions of universality of an n-point quadrature formula Q n are discussed. Universality means that Q n is able to compete with the respective optimal formula in many classes of functions. It is proved in a certain sense that the Gaus-sian quadrature formula satisses such a universality criterion. The underlying classes of functions are In each of these classes, we loose at most the ...
متن کاملOn an optimal quadrature formula in Sobolev space L2(m)(0, 1)
In this paper in the space L (m) 2 (0, 1) the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed N and for any m ≥ 2. It is ...
متن کامل