Best approximation of unbounded by bounded operators, and allied problems

Arestov, V. V.

doi:10.1007/bf01159122

Cited by 11 publications

(27 citation statements)

References 3 publications

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“…After Kolmogorov obtained his inequality (see [1][2][3]), many inequalities of this type were obtained for the norms of integer derivatives of univariate functions (see, e.g., [4][5][6][7][8]). …”

Section: Introduction Statement Of the Problem Main Resultsmentioning

confidence: 99%

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

2010

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Section: Introduction Statement Of the Problem Main Resultsmentioning

confidence: 99%

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

2010

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“…We fix α, β > 0 such that α + β = 1 / 2 and consider the class W ( α, β ) of functions x ∈ W satisfying conditions (8). It follows from the arguments presented above that…”

Section: Inequalities For Functions With Summable Second Derivativementioning

confidence: 99%

Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

2006

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“…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 semiaxis I = [0, ∞) with 0 ≤ k < n. In particular, Yu.…”

Section: The Problem and Its Historical Backgroundmentioning

confidence: 99%

“…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 semiaxis I = [0, ∞) with 0 ≤ k < n. In particular, Yu. N. Subbotin and L. V. Taikov [5] have solved the latter problem in the space L 2 (−∞, +∞) for arbitrary k and n, 0 < k < n. In the space L 2 (0, ∞) even in the case k = 1, n = 2 the exact solution to this problem, i.e., the solution to problem (2), is unknown.…”

Section: The Problem and Its Historical Backgroundmentioning

confidence: 99%

“…Problem (2) is connected not only with the exact inequality (3), but also with the optimal differentiation of functions (or the optimal restoration of a differentiation operator on functions) from the space W 2 2 (0, ∞) which are defined with some error in L 2 (0, ∞) (e.g., [1][2][3]14]). Let R be one of the following three sets of mappings from …”

Section: Evidently U (T ) ≤ U (T )mentioning

confidence: 99%

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Approximation of differentiation operator in the space L 2 on semiaxis

Arestov

Filatova

2013

Russ Math.

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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 *

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Best approximation of unbounded by bounded operators, and allied problems

Cited by 11 publications

References 3 publications

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Approximation of differentiation operator in the space L 2 on semiaxis

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