Polynomial approximation of operators, approximation accuracy bounds, and convergence theorems are highly important for both theoretical and applied analysis. The few published results in this area include the following: a generalization of the Weierstrass theorem to a separable Hilbert space [1] and to a Banach space [2]; the Pr6chet theorem on the limit of a sequence of integral polynomial functionals defined on the space of continuous functions [3]. and a generalization of this theorem to a Banach space [4]; construction of interpolation operator polynomials ir~ vector, Banach, and Hilbert spaces ~5-9]; some accuracy bounds and convergence theorems for polynomial operator interpolants [10][11][12].In this paper, we construct an operator interpolant for a polynomial operator on an orthonormal system of knots in a Hilbert space. For the construction of this interpolant, we generalize at the operator level the method of orthogonal moments, which is used in problems of identification of polynomial functional systems with subsequent application of the scheme for separation of homogeneous functions [13, 14]. Then the interpolation accuracy is estimated for the constructed interpolant; for the case when the orthonormal system of knots is a basis of the space, the accuracy bound leads to pointwise convergence of the operator interpolation process as the number of knots increases. Such a convergence example is considered for a regular functional polynomial of second degree defined on periodic functions whose derivatives are of bounded variation.Let X be a Hilbert space and Y a vector space. We define the set l
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 paper is a continuation of [1]. It generalizes the results connected with the Hermite operator interpolation in Hilbert spaces. In [1], the interpolational conditions at nodes are stated in terms of Gateaux differentials in repeating directions. Thus, the kth Gateaux differential at a node contains all k-1 directions involved in the (k -1)th differential at the same node, which, generally, restricts the range of applicability of the Hermite operator polynomial. In this paper this restriction is removed, and directions of differentiation can be chosen arbitrarily. Similarly to [1], the degree of the polynomial n, the number of nodes m, and orders ki, i = 1,..., m, of Gateaux differentials at nodes are not related with one another. lax this paper the following results are obtained. For any m, n, ki, and i = 1,... ,m, a necessary and sufficient condition for solvability of the problem of the Hermite operator interpolation in a Hilbert space is established. Under this condition the set of all Hermite operator polynomials and, also, the set of interpolants preserving operator polynomials of the same degree are described. Among the few publications in the field of operator interpolation, we should note [2], in which a Hermite operator polynomial mapping a space into itself is constructed. The author of [2] restricts himself to first Fr6chet derivatives and considers only the classical case n = 2m -1, while the interpolant itself does not preserve operator polynomials of the saane degree.
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