On optimal harmonic synthesis from inaccurate spectral data

Magaril-Il'yaev, Georgii Georgievich; Osipenko, K. Yu.

doi:10.1007/s10688-010-0029-7

Cited by 15 publications

(3 citation statements)

References 5 publications

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“…It turned out that in some cases it is possible to construct a whole family of optimal recovery methods for a linear operator. The study of such families began in [20] and continued in [21], [22], [14] and [19].…”

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confidence: 99%

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

Osipenko

2024

Izv. Math.

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The paper proposes an approach for construction of families of optimal methods for the recovery of linear operators from inaccurately given information. The proposed method of construction is then applied to recover derivatives from inaccurately specified other derivatives in the multidimensional case and to recover solutions of the heat equation from inaccurately specified temperature distributions at some time instants.

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confidence: 99%

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

Osipenko

2024

Izv. Math.

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“…The problems of this kind are also considered in [3]. Based on the general principles of extremal problems the new approach can be found in [5,13,7,6], as well as some results in this area. In the papers [8,4] authors obtained some inequalities for derivatives and showed that the problem of finding the exact constants in such inequalities can be formulated and efficiently solved as the corresponding optimal recovery problem.…”

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confidence: 99%

On a problem of optimal recovery and Kolmogorov type inequalities on an interval

Bagramyan

2015

Journal of Approximation Theory

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We consider an optimal recovery problem for the k-th derivative of the function on an interval from the information on the function itself, given in the mean square metric. As a consequence of the solution we prove one Kolmogorov type inequality for derivatives on an interval and demonstrate that the constant in this inequality can be reduced by considering particular subsets of the function class.Optimal recovery problem first appeared in the paper of Smolyak [15] and has been widely developed in [12,[9][10][11]. The problems of this kind are also considered in [3]. Based on the general principles of extremal problems the new approach can be found in [5,13,7,6], as well as some results in this area. In the papers [8,4] authors obtained some inequalities for derivatives and showed that the problem of finding the exact constants in such inequalities can be formulated and efficiently solved as the corresponding optimal recovery problem. In this paper we develop their approach and prove one Kolmogorov type inequality for derivatives (originally obtained in [14] and discussed in the paragraph 5.3 of the book [2]) as a consequence of the solution of the optimal recovery problem. Moreover, we show that the constant in this inequality, which

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“…Пусть ϕ a -такое отображение. Оценим квадрат максимизируемого функционала в (14) с ϕ = ϕ a , переходя по теореме Планшереля к образам Фурье (f (ξ) λ(A), если ξ ∈ R d \ A):…”

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О Наилучших Методах Восстановления Производных На Соболевских Классах

Магарил-Ильяев¹,

Magaril-Il'yaev²,

Осипенко³

et al. 2014

Известия Российской академии наук. Серия математическая

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О наилучших методах восстановления производных на соболевских классах Построены наилучшие (оптимальные) методы восстановления производных функций из обобщенного соболевского класса функций на R d при условии, что о каждой такой функции известно точно или приближенно ее преобразование Фурье на произвольном измеримом множестве A ⊂ R d. В обоих случаях построены семейства оптимальных методов. В этих методах используется не весь объем информации о преобразовании Фурье, однако та часть, которая используется, подвергается некоторой фильтрации. Рассмотрена задача о нахождении наилучшего множества для восстановления данной производной среди всех множеств фиксированной меры. Библиография: 19 наименований.

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On optimal harmonic synthesis from inaccurate spectral data

Abstract: The problem of optimal recovery of derivatives of functions on the real line from inaccurate Fourier transforms of these functions on a finite interval is studied. Optimal recovery methods different in data handling procedures are constructed.

Cited by 15 publications

References 5 publications

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

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

On a problem of optimal recovery and Kolmogorov type inequalities on an interval

О Наилучших Методах Восстановления Производных На Соболевских Классах

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