Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space L2

Arestov, V. V.

doi:10.1134/s0081543820020029

Cited by 4 publications

(7 citation statements)

References 19 publications

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“…Note some facts. Necessary and sufficient conditions for the finiteness of quantity (1.2) are known, see [26] (s = q) and [4] (s = q). Roughly speaking, these conditions are s ≥ r, q ≥ p.…”

Section: Stechkin's Problem On the Best Approximation Of Differentiat...mentioning

confidence: 99%

“…For a discussion of conditions (1.4), see [4]. Under conditions (1.3) and (1.4), we have the formula…”

Section: Stechkin's Problem On the Best Approximation Of Differentiat...mentioning

confidence: 99%

“…The following statement is contained in the author's paper [8, Theorem 3]; however, this result was preceded by several years of research by the author, see [4][5][6]8] and [13,14].…”

Section: Two Extremal Problems Related To Stechkin's Problem (12)mentioning

confidence: 99%

“…For even n ≥ 2 and 1 < r < ∞, the statements of Theorem 4, generally speaking, no longer hold. The author's paper [11] provides a solution to problem (3.33) for n = 2 (k = 1); r = s = 2, p = q = ∞.…”

Section: Case N = 2 R =mentioning

confidence: 99%

“…The following inequality holds [11,13] for the best constants in (3.39) and, in particular, in (3.40):…”

Section: Case N = 2 R =mentioning

confidence: 99%

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APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF \((p,q)\)-MULTIPLIERS AND THEIR PREDUAL SPACES

Arestov

2023

UMJ

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We consider a variant \(E_{n,k}(N;r,r;p,p)\) of the four-parameter Stechkin problem \(E_{n,k}(N;r,s;p,q)\) on the best approximation of differentiation operators of order \(k\) on the class of \(n\) times differentiable functions \((0<k<n)\) in Lebesgue spaces on the real axis. We discuss the state of research in this problem and related problems in the spaces of multipliers of Lebesgue spaces and their predual spaces. We give two-sided estimates for \(E_{n,k}(N;r,r;p,p)\). The paper is based on the author's talk at the S.B. Stechkin's International Workshop-Conference on Function Theory (Kyshtym, Chelyabinsk region, August 1–10, 2023).

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Section: Stechkin's Problem On the Best Approximation Of Differentiat...mentioning

confidence: 99%

“…For a discussion of conditions (1.4), see [4]. Under conditions (1.3) and (1.4), we have the formula…”

Section: Stechkin's Problem On the Best Approximation Of Differentiat...mentioning

confidence: 99%

“…The following statement is contained in the author's paper [8, Theorem 3]; however, this result was preceded by several years of research by the author, see [4][5][6]8] and [13,14].…”

Section: Two Extremal Problems Related To Stechkin's Problem (12)mentioning

confidence: 99%

Section: Case N = 2 R =mentioning

confidence: 99%

“…The following inequality holds [11,13] for the best constants in (3.39) and, in particular, in (3.40):…”

Section: Case N = 2 R =mentioning

confidence: 99%

See 3 more Smart Citations

APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF \((p,q)\)-MULTIPLIERS AND THEIR PREDUAL SPACES

Arestov

2023

UMJ

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Optimal recovery in weighted spaces with homogeneous weights

Osipenko¹

2022

Sb. Math.

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The paper concerns problems of the recovery of operators from noisy information in weighted -spaces with homogeneous weights. A number of general theorems are proved and applied to problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of the Laplace operator from a noisy Fourier transform in the -metric. Bibliography: 30 titles.

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On the construction of families of optimal recovery methods for linear operators

Osipenko

2024

Известия Российской академии наук. Серия математическая

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Предлагается некоторый подход к построению семейств оптимальных методов восстановления линейных операторов по неточно заданной информации. Предложенный метод построения применяется для восстановления производных по неточно заданным другим производным в многомерном случае и для восстановления решений уравнения теплопроводности по неточно заданным распределениям температур в некоторые моменты времени. Библиография: 25 наименований.

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Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space L2

Cited by 4 publications

References 19 publications

APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF \((p,q)\)-MULTIPLIERS AND THEIR PREDUAL SPACES

APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF \((p,q)\)-MULTIPLIERS AND THEIR PREDUAL SPACES

Optimal recovery in weighted spaces with homogeneous weights

On the construction of families of optimal recovery methods for linear operators

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