On interpolation of polynomial operators

Abstract

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

On interpolation approximation of differentiable operators in a Hilbert space

“…From the viewpoint of complexity and informativeness of a construction, the Hermite-type interpolation (4) occupies an intermediate place between Lagrange [3,7] and Newton [6,7] interpolation operator polynomials. For the construction of an interpolation formula of the Lagrange type, only information on the values of the operator at nodes is necessary.…”

“…In the present paper, we continue the investigations carried out in [1][2][3][4][5][6][7][8] and devoted to the construction of interpolation operator approximations in Hilbert spaces and to the examination of their accuracy. The problems of the construction of an operator polynomial of the Lagrange type on a specially selected set of nodes L(m) and of the analysis of the accuracy of interpolation of polynomial and entire operators were considered earlier in [4].…”

Interpolation and identification problems

“…Pupkov, Kapalin, and Yushchenko [7] addressed the identification of polynomial functional systems by the method of orthogonal moments followed by selection of homogeneous functions. Khlobystov and Kashpur [8] showed that this method is equivalent to the interpolation one on the corresponding set of nodes and considered its generalization to operators in a Hilbert space.…”

“…Let us determine the error between the system (1) and its mathematical model (2) in this metric. Accuracy of the Lagrange-type interpolation formula (7) in the metric H and in the metric of the linear normalized space Y are estimated in [8,12] for polynomial operators, …”