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.
\Bibitem{Luk04}
\by A.~L.~Lukashov
\paper Inequalities for derivatives of rational functions on several intervals
\jour Izv. Math.
\yr 2004
\vol 68
\issue 3
\pages 543--565
\mathnet{http://mi.mathnet.ru/eng/im488}
\crossref{https://doi.org/10.1070/IM2004v068n03ABEH000488}
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Linking options:
https://www.mathnet.ru/eng/im488
https://doi.org/10.1070/IM2004v068n03ABEH000488
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This publication is cited in the following 36 articles:
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Kalmykov S., Nagy B., Totik V., “Bernstein- and Markov-Type Inequalities For Rational Functions”, Acta Math., 219:1 (2017), 21–63
Akturk M.A., Lukashov A., “Sharp Markov-Type Inequalities For Rational Functions on Several Intervals”, J. Math. Anal. Appl., 436:2 (2016), 1017–1022
Lukashov A.L., Szabados J., “The order of Lebesgue constant of Lagrange interpolation on several intervals”, Period. Math. Hung., 72:2 (2016), 103–111
Ibrahimoglu B.A., Cuyt A., “Sharp Bounds for Lebesgue Constants of Barycentric Rational Interpolation at Equidistant Points”, Exp. Math., 25:3 (2016), 347–354
Alexey Lukashov, Dmitri Prokhorov, “Approximation of sgn $$(x)$$ ( x ) on Two Symmetric Intervals by Rational Functions with Fixed Poles”, Comput. Methods Funct. Theory, 2015
Totik V., “Bernstein- and Markov-Type Inequalities For Trigonometric Polynomials on General Sets”, Int. Math. Res. Notices, 2015, no. 11, 2986–3020
Béla Nagy, Vilmos Totik, “Riesz-type inequalities on general sets”, Journal of Mathematical Analysis and Applications, 2014
A. V. Olesov, “Inequalities for majorizing analytic functions and their applications to rational trigonometric functions and polynomials”, Sb. Math., 205:10 (2014), 1413–1441
S. I. Kalmykov, “On some rational functions which are analogues of Chebyshev polynomials”, J. Math. Sci. (N. Y.), 207:6 (2015), 874–884
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
Alexey Lukashov, Sergey Tyshkevich, “On trigonometric polynomials deviating least from zero on an interval”, Journal of Approximation Theory, 168 (2013), 18
Mehmet Akturk, Alexey Lukashov, “Weighted analogues of Bernstein-type inequalities on several intervals”, J Inequal Appl, 2013:1 (2013), 487
Vilmos Totik, Tamás Varga, “A sharp Lp-Bernstein inequality on finitely many intervals”, ActaSci.Math., 79:3-4 (2013), 401
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