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Uniform estimate for a segment function in terms of a polynomial strip

Authors: S. I. Dudov and E. V. Sorina
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 723-742
MSC (2010): Primary 90C05
Published electronically: July 24, 2013
MathSciNet review: 3087820
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Abstract | References | Similar Articles | Additional Information

Abstract: A problem about a uniform estimate for a continuous segment function in terms of a polynomial strip is considered. The problem reduces to a convex programming problem the target function of which is equal to the sum of the target functions for the inner and outer estimate of the same segment function via a polynomial strip. Convex analysis tools are used to obtain necessary and sufficient conditions for being a solution, and also uniqueness conditions in a form resembling the Chebyshov alternance.

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  • 1. B. N. Pšeničnyĭ, Vypuklyi analiz i ekstremalnye zadachi, “Nauka”, Moscow, 1980 (Russian). Seriya “Nelineinyi Analiz i ego Prilozheniya”. [Series in Nonlinear Analysis and its Applications]. MR 581125
  • 2. V. F. Dem′yanov and L. V. Vasil′ev, Nedifferentsiruemaya optimizatsiya, \cyr Optimizatsiya i Issledovanie Operatsiĭ. [Optimization and Operations Research], “Nauka”, Moscow, 1981 (Russian). MR 673171
    Vladimir F. Dem′yanov and Leonid V. Vasil′ev, Nondifferentiable optimization, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1985. Translated from the Russian by Tetsushi Sasagawa. MR 816531
  • 3. V. F. Dem′yanov and A. M. Rubinov, Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, \cyr Optimizatsiya i Issledovanie Operatsiĭ [Optimization and Operations Research], vol. 23, “Nauka”, Moscow, 1990 (Russian). MR 1120545
  • 4. E. S. Polovinkin, Theory of multivalued mappings, Moskov. Fiz.-Tekhn. Inst., Moscow, 1983. (Russian)
  • 5. G. G. Magaril-Il′yaev and V. M. Tikhomirov, Convex analysis: theory and applications, Translations of Mathematical Monographs, vol. 222, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Russian edition by Dmitry Chibisov and revised by the authors. MR 2013877
  • 6. Blagovest Sendov, Khausdorfovye priblizheniya, Bolgar. Akad. Nauk, Sofia, 1979 (Russian). MR 534426
  • 7. V. K. Dzjadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0612836
    Vladislav K. Dzyadyk and Igor A. Shevchuk, Theory of uniform approximation of functions by polynomials, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. Translated from the Russian by Dmitry V. Malyshev, Peter V. Malyshev and Vladimir V. Gorunovich. MR 2447076
  • 8. Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • 9. I. Yu. Vygodchikova, S. I. Dudov, and E. V. Sorina, Outer estimation of a segment function by a polynomial strip, Zh. Vychisl. Mat. Mat. Fiz. 49 (2009), no. 7, 1175–1183 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 49 (2009), no. 7, 1119–1127. MR 2599389, 10.1134/S0965542509070057
  • 10. F. L. Chernous′ko, Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem, “Nauka”, Moscow, 1988 (Russian). Metod ellipsoidov. [The method of ellipsoids]. MR 946472
  • 11. Alexander Kurzhanski and István Vályi, Ellipsoidal calculus for estimation and control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA; International Institute for Applied Systems Analysis, Laxenburg, 1997. MR 1419317
  • 12. M. S. Nikol′skiĭ, Approximation of a continuous multivalued mapping by constant multivalued mappings, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 1 (1990), 76–79, 82 (Russian); English transl., Moscow Univ. Comput. Math. Cybernet. 1 (1990), 73–76. MR 1051650
  • 13. S. I. Dudov and A. B. Konoplev, On the approximation of a continuous multivalued mapping by constant multivalued mappings with ball images, Mat. Zametki 82 (2007), no. 4, 525–529 (Russian, with Russian summary); English transl., Math. Notes 82 (2007), no. 3-4, 469–473. MR 2375788, 10.1134/S0001434607090222
  • 14. V. F. Demyanov and V. N. Malozemov, Vvedenie v minimaks, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0475822
    V. F. Dem′yanov and V. N. Malozemov, Introduction to minimax, Halsted Press [John Wiley & Sons], New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1974. Translated from the Russian by D. Louvish. MR 0475823

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Additional Information

S. I. Dudov
Affiliation: Saratov state university, Astrakhanskaya str. 83, Saratov 410012, Russia

E. V. Sorina
Affiliation: Saratov state university, Astrakhanskaya str. 83, Saratov 410012, Russia

Keywords: Segment function, polynomial strip, uniform estimate, subdifferential, alternance
Received by editor(s): September 9, 2011
Published electronically: July 24, 2013
Additional Notes: The author was supported by RFBR (grant no. 10-01-00270-a) and by the NSh grant NSh-4383.2010.1.
Article copyright: © Copyright 2013 American Mathematical Society