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 Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 1, Pages 56–68 (Mi timm779)

On the localization of singularities of the first kind for a function of bounded variation

A. L. Ageevab, T. V. Antonovab

a Ural Federal University
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Methods of the localization (detection) of discontinuities of the first kind for a function of bounded variation of one variable are constructed and investigated. We consider the problem of localizing discontinuities of a function that is noisy in the space $L_2(-\infty,+\infty)$. We distinguish between discontinuities with the absolute value of the jump greater than some positive $\Delta^{\min}$ and discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities and localize them using the approximately given function and the error level. Since the problem is ill-posed, regularizing algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The (order) optimality of the constructed methods on the classes of functions with singularities is established.

Keywords: ill-posed problem, discontinuity of the first kind, localization of singularities, regularizing method.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2013, 280, suppl. 1, 13–25

Bibliographic databases:

Document Type: Article
UDC: 517.988.68

Citation: A. L. Ageev, T. V. Antonova, “On the localization of singularities of the first kind for a function of bounded variation”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 1, 2012, 56–68; Proc. Steklov Inst. Math. (Suppl.), 280, suppl. 1 (2013), 13–25

Citation in format AMSBIB
\Bibitem{AgeAnt12} \by A.~L.~Ageev, T.~V.~Antonova \paper On the localization of singularities of the first kind for a~function of bounded variation \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2012 \vol 18 \issue 1 \pages 56--68 \mathnet{http://mi.mathnet.ru/timm779} \elib{http://elibrary.ru/item.asp?id=17358678} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2013 \vol 280 \issue , suppl. 1 \pages 13--25 \crossref{https://doi.org/10.1134/S0081543813020028} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000317236500002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84875968124} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. T. Faizullin, R. R. Faizullin, “Vosstanovlenie lineinykh funktsionalnykh zavisimostei s zadannoi osobennostyu”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:1 (2014), 103–108
2. A. L. Ageev, T. V. Antonova, “O diskretizatsii metodov lokalizatsii osobennostei zashumlennoi funktsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 3–13
3. A. L. Ageev, T. V. Antonova, “Methods for the approximating the discontinuity lines of a noisy function of two variables with countably many singularities”, J. Appl. Industr. Math., 9:3 (2015), 297–305
4. A. L. Ageev, T. V. Antonova, “Localization of boundaries for subsets of discontinuity points of noisy function”, Russian Math. (Iz. VUZ), 61:11 (2017), 10–15
5. A. L. Ageev, T. V. Antonova, “Otsenki kharakteristik metodov lokalizatsii razryvov pervogo roda zashumlennoi funktsii”, Sib. zhurn. industr. matem., 22:1 (2019), 3–12
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