On interpolation approximation of differentiable operators in a Hilbert space

Abstract

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.

Interpolation and identification problems

“…The present study refines the relationship between the method of orthogonal moments for identification of polynomial systems and the method of Lagrange-type operator interpolation of a special sequence of nodes and Hermitian-type operator interpolation [9] for differentiable operators in a Hilbert space. We will estimate interpolation accuracy using the identification problem in the L T 2 0 ( , )space and the minimum number of input signals providing prescribed accuracy.…”

“…Let us pass to the Hermite-type interpolation problem [9] for polynomial operators (4). On the assumption that …”

“…For polynomial operators, the authors proved in [9] the equivalence of the interpolation Lagrange information on the set of nodes w i N i , , = 0 , and the Hermite-type one (11) Let us pass to analyzing the accuracy of identification. Let X and Y be Hilbert spaces (X is separable), m be a Gaussian measure on X whose first moment is equal to zero, and the linear operator B X X : ® , where q e i i Î = ( , ), , , , .…”