On the accuracy of polynomial interpolation in Hilbert space with disturbed nodal values of the operator
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, \textit{V. V. Khlobystov} and \textit{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 \(\| F-\widetilde{P}_n^I\| _H\) 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.
scientific article; zbMATH DE number 1981993