Approximation of differentiation operator in the space L 2 on semiaxis

Abstract

Abstract-We establish an upper bound for the error of the best approximation of the first order differentiation operator by linear bounded operators on the set of twice differentiable functions in the space L 2 on the half-line. This upper bound is close to a known lower bound and improves the previously known upper bound due to E. E. Berdysheva. We use a specific operator that is introduced and studied in the paper. DOI: 10.3103/S1066369X13050010Keywords and phrases: Stechkin problem, optimal recovery, differential operator, half-line.1. The problem and its historical background. In this paper we consider the problem on the best approximation of a differential operator (of the first order) by linear bounded operators on the class of twice differentiable functions in the space L 2 = L 2 (0, ∞) of real-valued measurable functions f , whose square is summable on the semiaxis (0, ∞), which is equipped with the normMore precisely, let W 2 2 = W 2 2 (0, ∞) be the space of functions f ∈ L 2 (0, ∞), which are defined and continuously differentiable on [0, ∞), whose derivative f is locally absolutely continuous on the semiaxis [0, ∞), and the second derivative belongs to the space L 2 (0, ∞). In W 2 2 = W 2 2 (0, ∞) we extract the class Q 2 2 = Q 2 2 (0, ∞) of functions f such that f ≤ 1. In what follows, B = B 2 (0, ∞) is the set of linear bounded operators in the space L 2 (0, ∞), and B(N ) is the set of operators S ∈ B, whose norm is bounded by the number N > 0, i.e., S L 2 →L 2 ≤ N . For a concrete operator S ∈ B the valueis the deviation of the operator S from the differentiation operator in the space L 2 (0, ∞) on the class Q 2 2 . The problem under considerations consists in studying the valueof the best approximation in the space L 2 (0, ∞) on the class Q 2 2 of the differentiation operator by the set B(N ) of linear bounded operators, whose norms are bounded by the number N > 0.Problem (2) is a particular case of a more general problem on the best approximation of an unbounded linear operator by linear bounded ones on some class of elements; this problem was stated in 1967 by S. B. Stechkin [1]. Many papers [2][3][4] are dedicated to the Stechkin problem. This problem has been most completely studied for the differentiation operator of order k on the class of n times differentiable functions in spaces L p (I), 1 ≤ p ≤ ∞, on the numerical axis I = (−∞, ∞), and on the *

On the Best Approximation of the Infinitesimal Generator of a Contraction Semigroup in a Hilbert Space

“…Problem (3.1) was fully solved only in 2014 by Arestov and the second named author [3] . Namely, they showed that…”

“…Integrating by parts and taking into account that lim t→∞ y(t) = 0, we obtain (see [3] for details) that…”

“…This problem first appeared in Stechkin's work in 1965-1967 [11]. The problem was studied by a number of authors (see surveys [1], [2], monograph [4], paper [3], and the bibliography therein).…”