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 Mat. Zametki, 1998, Volume 63, Issue 1, Pages 69–80 (Mi mz1249)

On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle

A. S. Leonov

Moscow Engineering Physics Institute (State University)

Abstract: We introduce the new notion of total variation for the Hardy class of functions of several variables and state various properties, similar to those in the one-dimensional case, for functions belonging to this class. In particular, we prove a precise version of Helly's selection principle for this class.

DOI: https://doi.org/10.4213/mzm1249

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English version:
Mathematical Notes, 1998, 63:1, 61–71

Bibliographic databases:

UDC: 517.397

Citation: A. S. Leonov, “On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle”, Mat. Zametki, 63:1 (1998), 69–80; Math. Notes, 63:1 (1998), 61–71

Citation in format AMSBIB
\Bibitem{Leo98} \by A.~S.~Leonov \paper On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle \jour Mat. Zametki \yr 1998 \vol 63 \issue 1 \pages 69--80 \mathnet{http://mi.mathnet.ru/mz1249} \crossref{https://doi.org/10.4213/mzm1249} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1631844} \zmath{https://zbmath.org/?q=an:0924.26007} \transl \jour Math. Notes \yr 1998 \vol 63 \issue 1 \pages 61--71 \crossref{https://doi.org/10.1007/BF02316144} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000075520700008} 

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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. Leonov, AS, “Numerical piecewise-uniform regularization for two-dimensional ill-posed problems”, Inverse Problems, 15:5 (1999), 1165
2. A. S. Leonov, “Primenenie funktsii neskolkikh peremennykh s ogranichennymi variatsiyami dlya chislennogo resheniya dvumernykh nekorrektnykh zadach”, Sib. zhurn. vychisl. matem., 2:3 (1999), 257–271
3. Chistyakov, VV, “Superposition operators in the algebra of functions of two variables with finite total variation”, Monatshefte fur Mathematik, 137:2 (2002), 99
4. Balcerzak, M, “On Helly's principle for metric semigroup valued by mappings of two real variables”, Bulletin of the Australian Mathematical Society, 66:2 (2002), 245
5. V. V. Chistyakov, “Abstract superposition operators on mappings of bounded variation of two real variables. I”, Siberian Math. J., 46:3 (2005), 555–571
6. Chistyakov, VV, “A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. II”, Nonlinear Analysis-Theory Methods & Applications, 63:1 (2005), 1
7. Chistyakov, VV, “A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. I”, Nonlinear Analysis-Theory Methods & Applications, 62:3 (2005), 559
8. Kohl, N, “Evolving neural networks for strategic decision-making problems”, Neural Networks, 22:3 (2009), 326
9. Chistyakov V.V., Tretyachenko Yu.V., “Maps of Several Variables of Finite Total Variation. I. Mixed Differences and the Total Variation”, J. Math. Anal. Appl., 370:2 (2010), 672–686
10. Chistyakov V.V., Tretyachenko Yu.V., “Maps of Several Variables of Finite Total Variation. II. E. Helly-Type Pointwise Selection Principles”, J. Math. Anal. Appl., 369:1 (2010), 82–93
11. Chistyakov V.V., Tretyachenko Yu.V., “Selection Principles for Maps of Several Variables”, Dokl. Math., 81:2 (2010), 282–285
12. A. S. Leonov, “Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems”, Proc. Steklov Inst. Math. (Suppl.), 280, suppl. 1 (2013), 119–133
13. Bantsuri L.D., Oniani G.G., “On Differential Properties of Functions of Bounded Variation”, Anal. Math., 38:1 (2012), 1–17
14. Lind M., “Estimates of the Total P-Variation of Bivariate Functions”, J. Math. Anal. Appl., 401:1 (2013), 218–231
15. Chistyakov V.V., Tretyachenko Yu.V., “A Pointwise Selection Principle for Maps of Several Variables via the Total Joint Variation”, J. Math. Anal. Appl., 402:2 (2013), 648–659
16. Aziz W., Leiva H., Merentes N., “Solutions of Hammerstein Equations in the Space $BV (I_a^b)$”, Quaest. Math., 37:3 (2014), 359–370
17. Aistleitner Ch., Dick J., “Functions of Bounded Variation, Signed Measures, and a General Koksma-Hlawka Inequlity”, Acta Arith., 167:2 (2015), 143–171
18. Aistleitner Ch., Pausinger F., Svane A.M., Tichy R.F., “On functions of bounded variation”, Math. Proc. Camb. Philos. Soc., 162:3 (2017), 405–418
19. Chistyakov V.V., Chistyakova S.A., “Pointwise Selection Theorems For Metric Space Valued Bivariate Functions”, J. Math. Anal. Appl., 452:2 (2017), 970–989
20. Radulovic D., Wegkamp M., Zhao Yu., “Weak Convergence of Empirical Copula Processes Indexed By Functions”, Bernoulli, 23:4B (2017), 3346–3384
21. Bracamonte M., Ereu J., Gimenez J., Merentes N., “On Metric Semigroups-Valued Functions of Bounded Riesz-Phi-Variation in Several Variables”, Bol. Soc. Mat. Mex., 24:1 (2018), 133–153
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