Analysis of the Accuracy of Interpolation of Entire Operators in a Hilbert Space in the Case of Perturbed Nodal Values

Khlobystov, V. V.; Kashpur, E. F.

doi:10.1023/b:ukma.0000010612.26370.80

Cited by 2 publications

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“…. , e i k , and let T m,n (x) be the approximation of T n (x) given by (4) and satisfying the Hermite-type interpolation conditions (5). We give an estimate for the accuracy of approximation of the operator F (x) by the polynomial T m,n (x) in the metric of the space Y.…”

Section: Remarkmentioning

confidence: 99%

“…In the present paper, we continue the investigations carried out in [1][2][3][4][5][6][7][8] and devoted to the construction of interpolation operator approximations in Hilbert spaces and to the examination of their accuracy. The problems of the construction of an operator polynomial of the Lagrange type on a specially selected set of nodes L(m) and of the analysis of the accuracy of interpolation of polynomial and entire operators were considered earlier in [4].…”

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confidence: 96%

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On interpolation approximation of differentiable operators in a Hilbert space

Khlobystov

Popovicheva

2006

Ukr Math J

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In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives.In the present paper, we continue the investigations carried out in [1-8] and devoted to the construction of interpolation operator approximations in Hilbert spaces and to the examination of their accuracy. The problems of the construction of an operator polynomial of the Lagrange type on a specially selected set of nodes L(m) and of the analysis of the accuracy of interpolation of polynomial and entire operators were considered earlier in [4]. In the present paper, we propose an Hermite-type interpolation, which, as shown in what follows, is equivalent to the Lagrange interpolation on the set of nodes L(m) for polynomial operators and enables one, in the case of infinitely differentiable nonpolynomial operators, to strengthen the results concerning the convergence of interpolation processes (the case of Gâteaux analytic operators) and to obtain an estimate for the accuracy of approximations (the case of finitely many higher Fréchet derivatives of the operator), which is impossible within the framework of the Lagrange interpolation.Let X and Y be Hilbert spaces (X is a separable space) and let F : X → Y be a Gâteaux analytic operator in the space X.

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Section: Remarkmentioning

confidence: 99%

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confidence: 96%