Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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
DOI: https://doi.org/10.1090/S1061-0022-2013-01262-3
Published electronically: July 24, 2013
MathSciNet review: 3087820
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. B. N. Pshenichnyĭ, Convex analysis and extremal problems, Nauka, Moscow, 1980. (Russian) MR 0581125 (81k:90080)
  • 2. V. F. Dem'yanov and L. V. Vasil'ev, Nondifferentiable optimization, Nauka, Moscow, 1981; English transl., Optimization Software, Inc., Publ. Division, New York, 1985. MR 0673171 (84d:49002); MR 0816531 (86m:49002)
  • 3. V. F. Dem'yanov and A. M. Rubinov, Foundations of nonsmooth analysis, and quasidifferential calculus, Nauka, Moscow, 1990. (Russian) MR 1120545 (92j:40001)
  • 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 and its applications, Èditorial URSS, Moscow, 2000; English transl., Convex analysis: theory and applications, Transl. Math. Monogr., vol. 222, Amer. Math. Soc., Providence, RI, 2003. MR 2013877 (2004k:49002)
  • 6. Bl. Sendov, Hausdorff approximations, Bolgar. Akad. Nauk, Sofia, 1979. (Russian) MR 0534426 (80j:41004)
  • 7. V. K. Dzyadyk, Introduction to the theory of uniform approximation of functions by polynomials, Nauka, Moscow, 1977; English transl., Theory of uniform approximation of functions by polynomials, Walter de Gruyter, Berlin, 2008. MR 0612836 (58:29579); MR 2447076 (2009f:30001)
  • 8. S. Karlin and W. Studden, Tchebycheff systems: with applications in analysis and statistics, Pure Appl. Math., vol. 15, Intersci. Publ., New York etc., 1966. MR 0204922 (34:4757)
  • 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; English transl., Comput. Math. Math. Phys. 49 (2009), no. 7, 1119-1127. MR 2599389 (2011b:41042)
  • 10. F. L. Chernous'ko, Estimation of the phase state of dynamical systems: the method of ellipsoids, Nauka, Moscow, 1988. (Russian) MR 0946472 (89k:49028)
  • 11. A. B. Kurzhanski and I. Vályi, Ellipsoidal calculus for estimation and control, Systems and Control: Foundations and Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1419317 (98b:93003)
  • 12. M. S. Nikol'skiĭ, Approximation of a continuous multivalued mapping by constant multivalued mappings, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 1990, no. 1, 76-79; English transl., Moscow Univ. Comput. Math. Cybernet. 1990, no. 1, 73-76. MR 1051650 (91g:65160)
  • 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; English transl., Math. Notes 82 (2007), no. 3-4, 469-473. MR 2375788 (2009c:49038)
  • 14. V. F. Dem'yanov and V. N. Malozemov, Introduction to minimax, Nauka, Moscow, 1972; English transl., Halsted Press, New York-Toronto, 1974. MR 0475822 (57:15407a); MR 0475823 (57:15407b)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 90C05

Retrieve articles in all journals with MSC (2010): 90C05


Additional Information

S. I. Dudov
Affiliation: Saratov state university, Astrakhanskaya str. 83, Saratov 410012, Russia
Email: DudovSI@info.sgu.ru

E. V. Sorina
Affiliation: Saratov state university, Astrakhanskaya str. 83, Saratov 410012, Russia
Email: sorina@rol.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01262-3
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

American Mathematical Society