Interpolation of polynomial operators in a Hilbert space

Abstract

OPERATORS IN A HILBERTAn analysis of the accuracy of approximations of nonlinear operators in abstract spaces is of interest in both theoretical and applied mathematics. Among the few publications in this research field, we note the following papers. In [1], an extension of the Weierstrass theorem to a separable Hilber ~ space is obtained, while in [2] a theorem is given on the convergence of the interpolational polynomial process in a Banach space for abstract functions with a specially chosen sequence of nodes from R1. In [3], a theorem is proved on the convergence of a Newton-type interpolational operator process in a sphere for integer operators in a Banach space. In [4], the convergence of a Hermite-type interpolational process for polynomial operators is studied in a separable Hilbert measure space as the number of nodes increases, and an estimate for the rate of convergence is derived. In this case, the presence of the Gs differentials in the interpolational conditions as well as the dependence of an accuracy estimate on the correlation measure operator make the application of such a process in practice more difficult. In many applied problems, it is more advisable to use an interpolant with interpolational conditions containing no Gs differentials, with an accuracy estimate and convergence in the metric of the space of operator values. It is just such a situation to which this paper is devoted. Such research finds application in solving problems of identification of polynomial systems.Let X be a Hilbert space and Y be a vector space; let Fn: X --+ Y,

The numerical approximation of nonlinear functionals and functional differential equations

“…To this end, we can use the method for finding orthonormal moments of regular polynomial functionals (Theorem 1 in [102]). This yields…”

“…After n iterations, we have available all coefficients to construct the polynomial functional (155). It was shown in [102] (Theorem 2) that the following error estimate holds…”