Optimal Recovery of Elements of a Hilbert Space and their Scalar Products According to the Fourier Coefficients Known wi
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OPTIMAL RECOVERY OF ELEMENTS OF A HILBERT SPACE AND THEIR SCALAR PRODUCTS ACCORDING TO THE FOURIER COEFFICIENTS KNOWN WITH ERRORS V. F. Babenko,1 M. S. Gunko,2,3 and N. V. Parfinovych4
UDC 517.5
In a Hilbert space defined as the image of a unit ball under the action of a compact operator, we solve the problems of optimal recovery of elements according to their first n Fourier coefficients known with errors. Similar problems are also solved for the scalar products of elements from two different classes.
1. Introduction Suppose that a Banach space X, a class of elements W ⇢ X, an (information) set Y, and an (information) mapping I : W ! P0 (Y ), where P0 (Y ) is a collection of nonempty subsets of the set Y, are given. Assume that if we want to get information about an element x, then we receive an element of the set I(x). An arbitrary mapping Φ : Y ! X is called a method for the recovery of elements from the set W according to the available information. An error of the method of recovery in the class W according to information I is defined as the quantity E(W, I, Φ) = sup kx − Φ(y)kX .
(1)
E(W ; I) = inf E(W, I, Φ)
(2)
x2W y2I(x)
The quantity Φ
is called the error of optimal recovery of elements from the class W according to the data I. Moreover, the method Φ⇤ that realizes the greatest lower bound in (2) is called optimal. Let H1 and H2 be complex Hilbert spaces with scalar products h·, ·iH1 and h·, ·iH2 and norms k · kH1 and k · kH2 , respectively, and let A be a compact operator acting from H1 into H2 . By W A we denote the image of a unit ball of the space H1 under the action of the operator A, i.e., � W A = Ah : h 2 H1 , khkH1 1 .
Consider the problem of recovery of elements from the class W A in the case where X = H2 and Y = Cn . For an element x 2 W A , we know n its Fourier coefficients in a certain orthonormal system connected with the 1
Honchar Dnipro National University, Dnipro, Ukraine; e-mail: [email protected]. Honchar Dnipro National University, Dnipro, Ukraine; e-mail: [email protected]. 3 Corresponding author. 4 Honchar Dnipro National University, Dnipro, Ukraine; e-mail: [email protected]. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 736–750, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.1107. Original article submitted October 23, 2019. 0041-5995/20/7206–0853
c 2020
Springer Science+Business Media, LLC
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V. F. BABENKO , M. S. G UNKO ,
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AND
N. V. PARFINOVYCH
operator A with certain errors. The results obtained in the present paper complement and generalize the results on the recovery of functions obtained in [6]. The problem of recovery of linear operators in Hilbert spaces in the presence of exact information was studied in [1]. In the case where the information mapping I has the form Ix = i(x) + B, where i is a linear operator and B is a ball of certain radius (specifying the error), the corresponding recovery problem was considered in [2] (see also [3–5]). Another approach to the analysis of problems of this kind (base
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