On the problem of Hermite interpolation of operators in a Hilbert space

Abstract

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.

Interpolation of polynomial operators in a Hilbert space

“…If {xi}~__ 1 is a basis in X, then era(z) -~ 0 as m --+ oo for all x E X and the pointwise convergence of P~(x) to F,(x) as m --~ ee follows directly from estimate (6) [7], the description of the whole set of interpolational operator polynomials of a given degree in a Hilbert space over a fixed system of nodes is given. If a free polynomial appearing in that description is chosen to be pI(x), then P~(x) becomes an element of the set described.…”

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

“…In Hilbert spaces the interpolation problem can be solved, for instance, by a sort of generalization of the Lagrange formula (see [7]) constructed with scalar products, which is a particular case of polynomial operator interpolant ( [5,6], see [9]). This interpolant can be modified so to get a 112 G. Allasia and C. Bracco cardinal basis solution to the same problem (see [1]), obtaining acceptable approximation performances.…”