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optimal error of analytic continuation from a finite set with inaccurate data in hilbert spaces of holomorphic functions: Optimal error of analytic continuation from a finite set with inaccurate data in Hilbert spaces of holomorphic functions

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Bibliographic Details
Title: optimal error of analytic continuation from a finite set with inaccurate data in hilbert spaces of holomorphic functions: Optimal error of analytic continuation from a finite set with inaccurate data in Hilbert spaces of holomorphic functions
Authors: L. S. Maergoiz, A. M. Fedotov
Source: Siberian Mathematical Journal. 42(5):926-935
Publisher Information: Russian Academy of Sciences - RAS (Rossiĭskaya Akademiya Nauk - RAN), Siberian Branch (Sibirskoe Otdelenie), Sobolev Insitute of Mathematics (Institut Matematiki Im. S. L. Soboleva), Novosibirsk, 2001.
Publication Year: 2001
Subject Terms: analytic continuation, Entire functions of several complex variables, Removable singularities in several complex variables, Continuation of analytic objects in several complex variables, extrapolation with inaccurate data, optimal error, optimal linear algorithm
Description: The article under review refers to the papers [\textit{L.~S.~Maergojz}, Sib. Math. J. 41, No. 6, 1126-1136 (2000; Zbl 0970.32011) and Dokl. Math. 56, No. 2, 674-678 (1997; Zbl 0973.32002)] wherein the problem of optimal extrapolation from a finite set is studied in the class of entire functions with finite spectrum. The aim of the present article is to study this problem in the case of analytic continuation from a finite set with inaccurate data. To estimate the error of a linear functional, an approach is employed suggested by \textit{K.~Miller} in [SIAM J. Math. Anal. 1, 52-74 (1970; Zbl 0214.14804)] based on using the least squares method for ill-posed problems with a prescribed bound. As a result, the authors obtain constructive formulas for calculating the optimal error of the optimal linear algorithm for extrapolation from a set \(U\) to a point \(z_0\) in the class of functions \[ V = \{f\in H(D): \|f\|\leq r\}, \quad r > 0, \] where \(H\) is a Hilbert space with reproducing kernel. Moreover, the asymptotic behavior of the optimal error is investigated in the case when the errors of estimating the initial data tend to zero.
Document Type: Article
File Description: application/xml
ISSN: 0037-4466
DOI: 10.1023/a:1011967711386
Access URL: https://zbmath.org/1746519
https://link.springer.com/article/10.1023%2FA%3A1011967711386
Accession Number: edsair.dedup.wf.002..a40bceb4194ec2e42a7c72c4b51f7966
Database: OpenAIRE
Description
Abstract:The article under review refers to the papers [\textit{L.~S.~Maergojz}, Sib. Math. J. 41, No. 6, 1126-1136 (2000; Zbl 0970.32011) and Dokl. Math. 56, No. 2, 674-678 (1997; Zbl 0973.32002)] wherein the problem of optimal extrapolation from a finite set is studied in the class of entire functions with finite spectrum. The aim of the present article is to study this problem in the case of analytic continuation from a finite set with inaccurate data. To estimate the error of a linear functional, an approach is employed suggested by \textit{K.~Miller} in [SIAM J. Math. Anal. 1, 52-74 (1970; Zbl 0214.14804)] based on using the least squares method for ill-posed problems with a prescribed bound. As a result, the authors obtain constructive formulas for calculating the optimal error of the optimal linear algorithm for extrapolation from a set \(U\) to a point \(z_0\) in the class of functions \[ V = \{f\in H(D): \|f\|\leq r\}, \quad r > 0, \] where \(H\) is a Hilbert space with reproducing kernel. Moreover, the asymptotic behavior of the optimal error is investigated in the case when the errors of estimating the initial data tend to zero.
ISSN:00374466
DOI:10.1023/a:1011967711386