Optimal Recovery of the Operators of the Divided Difference of the Inaccurately Given Sequence by the Fourier Transform

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

Home Editorial board Publication ethics Peer review guidelines Current Archive Rules for authors Contacts Address: Markusa st. 22, Vladikavkaz, 362027, RNO-A, Russia Phone: (8672)50-18-06 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2018.3.18033 Optimal Recovery of the Operators of the Divided Difference of the Inaccurately Given Sequence by the Fourier Transform Unuchek S. A. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3. Abstract: In various applications, it is often necessary to reconstruct some characteristic of an object from some information (usually incomplete or inaccurate) about its other characteristics. There are various approaches to solving similar problems. In this paper, we used an approach based on the ideas of Andrei Nikolaevich Kolmogorov concerning the best means of approximation by finite-dimensional subspaces. The essence of the method lies in the fact that the best means of approximation on the whole class is sought. We consider the problem of simultaneous recovery of operators of divided differences of a sequence of all orders from 1 to \((n-1)\)th inclusive, in a class of sequences with bounded \(n\)th divided difference. The Fourier transform of this sequence is known inaccurately at a certain interval sequence in the mean square norm. A family of optimal recovery methods is constructed. Among the methods found are those that use minimal sequence information, pre-smoothing it. The exact value of the optimal error of recovering divided-difference operators is found. The passage to the limit from the obtained results implies a continuous case. Keywords: optimal recovery, operator of a divided difference, Fourier transform. Language: Russian Download the full text For citation: Unuchek S. A. Optimal Recovery of the Operators of the Divided Difference of the Inaccurately Given Sequence by the Fourier Transform, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 2018, vol. 20, no. 2, pp. 94-104. DOI 10.23671/VNC.2018.3.18033 + References 1. Smolyak S. A. On Optimal Recovery of Fuctions and Functionals of them, Candidate dissertation, Moscow State University, 1965 (in Russian). 2. Magaril-Il'yaev G. G., Osipenko K. Y. Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives, Functional Analysis and Its Applications, 2003, vol. 37, no. 3, pp. 203-214. DOI: 10.1023/A:1026084617039 3. Magaril-Il'yaev G. G, Osipenko K. Yu. Optimal Recovery of Operators from Inaccurate Information, Matematicheskij forum. T. $2$. Issledovanija po vypuklomu analizu. Itogi nauki. Juzhnyj federal'nyj okrug [Mathematical Forum, vol. 2, Studies on the Convex Analysis. Review of Science: Southern Federal District], 2009, vol. 2, pp. 158-192 (in Russian). 4. Unuchek S. A. Optimal Reconstruction of Divided Differences from an Inaccurate Sequence, Differential Equations, 2015, vol. 51, no. 7, pp. 948-954. DOI: 10.1134/S0012266115070125. 5. Unuchek S. A. On Optimal Recovery of the Operator of \(k\)-th Divided Difference from its Inaccurately Given Fourier Transfor, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Mathematical Journal], 2015, vol. 17, no. 3, pp. 84-92 (in Russian). DOI: 10.23671/VNC.2017.3.7268. 6. Magaril-Il'yaev G. G, Osipenko K. Yu. How Best to Recover a Function from its Inaccurately Given Spectrum? Mathematical Notes, 2012, vol. 92, no. 1, pp. 51-58. DOI: 10.1134/S0001434612070061. ← Contents of issue


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