The paper refines the relationship between the method of orthogonal moments for the identification of polynomial systems and the method of Lagrange-and Hermite-type operator interpolation in Hilbert space. The identification accuracy is estimated by the interpolation method and the minimum number of input signals that guarantee the prescribed accuracy is determined.Analysis of the properties and peculiarities of plants using modern information processing methods involves development of a mathematical model of a plant based on observable data -inputs and outputs. Such an approach is called plant identification and consists in determining structure and parameters of a plant. Moreover, many scientific and applied areas (medicine, economy, meteorology, seismology, hydro-and radiolocation, image processing, etc.) deal with various systems whose mathematical models can be represented by operator polynomials of some degree. This explains the importance and necessity of developing both theoretical and practical aspects of the analysis of polynomial plants. Among studies in this field, noteworthy are [1-3] concerned with data interpolation, system identification, and the theory of polynomial systems. Note that the possibility of approximating an arbitrary continuous operator by a polynomial one is justified by theorems similar to the Weierstrass theorem (approximation of a continuous function by polynomials). For example, the Frechet theorem is an analog of the Weierstrass theorem for a continuous functional. Prenter [4] has proved a theorem similar to the Weierstrass theorem for continuous nonlinear operators in a Hilbert space. Istratesku [5] and Daugavet [6] generalized this result to Banach space.The identification problem for plants of unknown structure may be considered as a problem of approximating operators in some functional spaces. Since operator interpolation is one of the methods of solving the approximation problem, it is important to establish relationships between operator interpolation formulas and identification methods for nonlinear systems under deterministic actions. 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.The present study refines the relationship between the method of orthogonal moments for identification of polynomial systems and the method of Lagrange-type operator interpolation of a special sequence of nodes and Hermitian-type operator interpolation [9] for differentiable operators in a Hilbert space. We will estimate interpolation accuracy using the identification problem in the L T 2 0 ( , )space and the minimum number of input signals providing prescribed accuracy. Let us consider the problem of identifying a nonlinear plant using the method of orthogonal moments fol...
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
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