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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 3, Pages 115–138 (Mi izv488)  

This article is cited in 30 scientific papers (total in 30 papers)

Inequalities for derivatives of rational functions on several intervals

A. L. Lukashov

Saratov State University named after N. G. Chernyshevsky

Abstract: We describe a solution of the problem of finding rational trigonometric functions with fixed denominator that deviate least from zero on several subintervals of the period. The resulting representation is used to prove inequalities that estimate the derivatives of rational trigonometric and algebraic functions with fixed denominator in terms of their values on several intervals. Particular cases of these inequalities include the well-known inequalities of Videnskii, Rusak, Totik and others.

DOI: https://doi.org/10.4213/im488

Full text: PDF file (1954 kB)
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English version:
Izvestiya: Mathematics, 2004, 68:3, 543–565

Bibliographic databases:

UDC: 517.5
MSC: 41A20, 41A50
Received: 15.12.2002

Citation: A. L. Lukashov, “Inequalities for derivatives of rational functions on several intervals”, Izv. RAN. Ser. Mat., 68:3 (2004), 115–138; Izv. Math., 68:3 (2004), 543–565

Citation in format AMSBIB
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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. Lukashov A.L., Peherstorfer F., “Zeros of polynomials orthogonal on two arcs of the unit circle”, J. Approx. Theory, 132:1 (2005), 42–71  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. L. Lukashov, “Ratsionalnye interpolyatsionnye protsessy na neskolkikh otrezkakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 5:1-2 (2005), 34–48  mathnet
    3. V. N. Dubinin, S. I. Kalmykov, “A majoration principle for meromorphic functions”, Sb. Math., 198:12 (2007), 1737–1745  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. L. Lukashov, “Estimates for derivatives of rational functions and the fourth Zolotarev problem”, St. Petersburg Math. J., 19:2 (2008), 253–259  mathnet  crossref  mathscinet  zmath  isi
    5. S. V. Tyshkevich, “On Chebyshev Polynomials on Arcs of a Circle”, Math. Notes, 81:6 (2007), 851–853  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. S. I. Kalmykov, “Majoration principles and some inequalities for polynomials and rational functions with prescribed poles”, J. Math. Sci. (N. Y.), 157:4 (2009), 623–631  mathnet  crossref  zmath
    7. V. N. Dubinin, D. B. Karp, V. A. Shlyk, “Izbrannye zadachi geometricheskoi teorii funktsii i teorii potentsiala”, Dalnevost. matem. zhurn., 8:1 (2008), 46–95  mathnet  elib
    8. V. N. Dubinin, “Emkosti kondensatorov i printsipy mazhoratsii v geometricheskoi teorii funktsii kompleksnogo peremennogo [Itogovyi nauchnyi otchet po mezhdistsiplinarnomu integratsionnomu proektu SO RAN: “Razrabotka teorii i vychislitelnoi tekhnologii resheniya obratnykh i ekstremalnykh zadach s prilozheniem v matematicheskoi fizike i gravimagnitorazvedke”]”, Sib. elektron. matem. izv., 5 (2008), 465–482  mathnet  mathscinet
    9. Lukashov A.L., Tyshkevich S.V., “Extremal polynomials on arcs of the circle with zeros on these arcs”, J. Contemp. Math. Anal., Armen. Acad. Sci., 44:3 (2009), 172–179  crossref  mathscinet  zmath  isi  elib  scopus
    10. Maergoiz L. S., Rybakova N. N., “Chebyshev polynomials with zeros lying on a circular arc”, Dokl. Math., 79:3 (2009), 319–321  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. Arestov V. V., “On trigonometric polynomials least deviating from zero”, Dokl. Math., 79:2 (2009), 280–283  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    12. A. L. Lukashov, S. V. Tyshkevich, “Ekstremalnye ratsionalnye funktsii na dugakh okruzhnosti s nulyami na etikh dugakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 8–13  mathnet  elib
    13. Arestov V.V., Mendelev A.S., “Trigonometric polynomials of least deviation from zero in measure and related problems”, Journal of Approximation Theory, 162:10 (2010), 1852–1878  crossref  mathscinet  zmath  isi  elib  scopus
    14. V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67:4 (2012), 599–684  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    15. Lukashov A., Akturk M.A., “Remez Type Inequality for Trigonometric Polynomials on an Interval”, First International Conference on Analysis and Applied Mathematics (ICAAM 2012), AIP Conference Proceedings, 1470, eds. Ashyralyev A., Lukashov A., Amer Inst Physics, 2012, 42–44  crossref  adsnasa  isi  scopus
    16. Alexey Lukashov, Sergey Tyshkevich, “On trigonometric polynomials deviating least from zero on an interval”, Journal of Approximation Theory, 168 (2013), 18  crossref  mathscinet  zmath  isi  scopus
    17. Mehmet Akturk, Alexey Lukashov, “Weighted analogues of Bernstein-type inequalities on several intervals”, J Inequal Appl, 2013:1 (2013), 487  crossref  mathscinet  zmath  isi  scopus
    18. Béla Nagy, Vilmos Totik, “Riesz-type inequalities on general sets”, Journal of Mathematical Analysis and Applications, 2014  crossref  mathscinet  isi  scopus
    19. A. V. Olesov, “Inequalities for majorizing analytic functions and their applications to rational trigonometric functions and polynomials”, Sb. Math., 205:10 (2014), 1413–1441  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. S. I. Kalmykov, “On some rational functions which are analogues of Chebyshev polynomials”, J. Math. Sci. (N. Y.), 207:6 (2015), 874–884  mathnet  crossref
    21. Akturk M.A., Lukashov A., “Markov-Type Inequalities For Rational Functions on Several Intervals”, International Conference on Analysis and Applied Mathematics, AIP Conference Proceedings, 1611, eds. Ashyralyev A., Malkowsky E., Amer Inst Physics, 2014, 208–210  crossref  isi  scopus
    22. Alexey Lukashov, Dmitri Prokhorov, “Approximation of sgn
      $$(x)$$
      ( x ) on Two Symmetric Intervals by Rational Functions with Fixed Poles”, Comput. Methods Funct. Theory, 2015  crossref  mathscinet  scopus
    23. Totik V., “Bernstein- and Markov-Type Inequalities For Trigonometric Polynomials on General Sets”, Int. Math. Res. Notices, 2015, no. 11, 2986–3020  crossref  mathscinet  zmath  isi  elib  scopus
    24. Akturk M.A., Lukashov A., “Sharp Markov-Type Inequalities For Rational Functions on Several Intervals”, J. Math. Anal. Appl., 436:2 (2016), 1017–1022  crossref  mathscinet  zmath  isi  scopus
    25. Lukashov A.L., Szabados J., “The order of Lebesgue constant of Lagrange interpolation on several intervals”, Period. Math. Hung., 72:2 (2016), 103–111  crossref  mathscinet  zmath  isi  elib  scopus
    26. Ibrahimoglu B.A., Cuyt A., “Sharp Bounds for Lebesgue Constants of Barycentric Rational Interpolation at Equidistant Points”, Exp. Math., 25:3 (2016), 347–354  crossref  mathscinet  zmath  isi  scopus
    27. Kalmykov S., Nagy B., Totik V., “Bernstein- and Markov-Type Inequalities For Rational Functions”, Acta Math., 219:1 (2017), 21–63  crossref  mathscinet  zmath  isi  scopus
    28. Kalmykov S., Nagy B., “Higher Markov and Bernstein Inequalities and Fast Decreasing Polynomials With Prescribed Zeros”, J. Approx. Theory, 226 (2018), 34–59  crossref  mathscinet  zmath  isi  scopus
    29. B. Eichinger, P. Yuditskii, “Ahlfors problem for polynomials”, Sb. Math., 209:3 (2018), 320–351  mathnet  crossref  crossref  adsnasa  isi  elib
    30. E. B. Bairamov, “Mnogochleny, naimenee uklonyayuschikhsya ot nulya na kvadrate kompleksnoi ploskosti”, Tr. IMM UrO RAN, 24, no. 3, 2018, 5–15  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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