@article {IOPORT.01981993,
    author       = {Khlobystov, V. V. and Kashpur, E. F.},
    title        = {On the accuracy of polynomial interpolation in Hilbert space with disturbed nodal values of the operator.},
    year         = {2002},
    journal      = {Cybernetics and Systems Analysis},
    volume       = {38},
    number       = {1},
    issn         = {1060-0396},
    pages        = {143-148 (2002); translation from Kibern. Sist. Anal. 2002, No. 1, 168-173},
    publisher    = {Springer (Consultants Bureau), New York, NY},
    doi          = {10.1023/A:1015560619575},
    abstract     = {Let $X$ be a separable Hilbert space, $Y$ a pre-Hilbert space, and $\mu$ a Gaussian measure on $X$ whose first moment is equal to 0. Then the correlation operator $B$ of $\mu$ is nuclear. Using $\mu$, it is possible to define a scalar product on $\Pi_n$, the space of all continuous polynomials from $X$ into $Y$ of degree $ \leq n$. An operator $F$ from $X$ into $Y$ is given by its values $F(Bx_i)$, $i = 0,\ldots,N$, where the $x_i$'s depend on the eigenvalues $\lambda_k$ and the associated orthonormal system of eigenvectors $e_k$ of $B$.\newline The authors consider the following problem. Find a polynomial $P_n \in \Pi_n$ with $P_n(Bx_i)=F(Bx_i)$ for all $i = 0,\ldots,N$. A solution $P_n^I$ can be constructed by the method of orthogonal moments, see, for example, {\it V. V. Khlobystov} and {\it E. F. Kashpur} [Cybern. Syst. Anal. 32, No. 3, 398--403 (1996; Zbl 0887.41005)]. In Theorem 1 it is given an error estimate of $\Vert F-\widetilde{P}_n^I\Vert _H$ in terms of the eigenvalues of $B$ and the accuracy $\delta$ of the values of $\widetilde{F}$ in the nodes. Theorem 2 shows that in the special case of $\lambda_k=q^k$ $(0<q<1)$ there is a natural number $m_0$ such that the interpolation accuracy cannot be improved if more than $m_0$ of the largest eigenvalues of $B$ are considered.},
    reviewer     = {Hans-Andreas Braun{\ss} (Potsdam)},
    identifier   = {01981993},
}