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In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m?2 and e ?x . Also some numerical results are presented. 相似文献
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This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the 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 nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P m?1(x), where P m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results. 相似文献
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The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_2~((m,0)) by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_2~((m,0)) is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue Dm(hβ) of the differential operator (d~(2m))/ (dx~(2m))-1 is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space W_2~((m,0)), which based on the discrete analogue D_m(hβ), is described. Next, for m = 1 and m = 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_2~((m,0)) for the cases m = 1 and m = 3 are presented. 相似文献
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Nurali Boltaev Abdullo Hayotov Gradimir Milovanovi Kholmat Shadimetov 《Journal of Applied Analysis & Computation》2017,7(4):1233-1266
This paper studies the problem of construction of optimal
quadrature formulas in the sense of Sard in the
$W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we
obtain new optimal quadrature formulas of such type for $N 1\geq
m$, where $N 1$ is the number of the nodes. Moreover, explicit
formulas for the optimal coefficients are obtained. We investigate
the order of convergence of the optimal formula for $m=1$. The obtained optimal quadrature formula in the
$W_2^{(m,m-1)}[0,1]$ space is exact for $\exp(-x)$ and
$P_{m-2}(x)$, where $P_{m-2}(x)$ is a polynomial of degree $m-2$.
Furthermore, we present some numerical results, which confirm the obtained theoretical results. 相似文献
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Abdullo Rakhmonovich Hayotov 《Lithuanian Mathematical Journal》2014,54(3):290-307
We construct a discrete analogue D m (hβ) of the differential operator d2m /dx 2m + 2ω 2d2m?2 /dx 2m?2 + ω 4d2m?4 /dx 2m?4 for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K 2(P 2) the discrete analogue D 2(hβ) for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are exact for trigonometric functions sin ωx and cos ωx. 相似文献
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Abdullo Rakhmonovich Hayotov Gradimir V. Milovanović Kholmat Mahkambaevich Shadimetov 《Numerical Algorithms》2011,57(4):487-510
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 L2(2)(0,1)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. 相似文献
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