On sharp Remez-type inequalities of various metrics for periodic functions

V.A. Kofanov

doi:10.15421/241609

On sharp Remez-type inequalities of various metrics for periodic functions

V.A. Kofanov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0003-0392-2257

Abstract

We prove new sharp Remez-type inequalities of various metrics for periodic functions, polynomials and splines.

Keywords

Remez-type inequalities; the comparison theorem; rearrangements

Full Text:

PDF (Русский)

References

Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280. doi:10.1007/BF02791137

Kofanov V.A. "Exact upper bounds of norms of functions and their derivatives on the classes of functions with given comparison function", Ukrainian Math. J., 2011; 63(7): pp. 969-984. (in Russian) doi:10.1007/s11253-011-0567-z

Remes E. "Sur une propriete еxtremale des polynomes de Tchebychef", Zapiski naukovo-doslid. in-tu matematyky i mekhaniky ta Kharkiv. mat. tov-va Kharkiv. derzh. un-tu. Seriya 4, 1936; 13(1): pp. 93-95. (in French)

Ganzburg M.I. "On a Remez-type inequality for trigonometric polynomials", J. of Approximation Theory, 2012; 164: pp. 1233-1237. doi:10.1016/j.jat.2012.05.006

Nursultanov E., Tikhonov S. "A sharp Remez inequality for trigonometric polynomials", Constructive Approximation, 2013; 38: pp. 101-132. doi:10.1007/s00365-012-9172-0

Ganzburg M.I. "Polynomial inequalities on measurable sets and their applications", Constructive Approximation, 2001; 17: pp. 275-306. doi:10.1007/s003650010020

Kofanov V.A. "Sharp Remez-type inequalities for differentiable periodic functions, polynomials and splines", Ukrainian Math. J., 2016; 68(2): pp. 227-240. (in Russian) doi:10.1007/s11253-016-1222-5

Pinkus A., Shisha O. "Variations on the Chebyshev and $$$L^q$$$-Theories of Best Approximation", Journal of Approx. Theory, 1982; 35(2): pp. 148-168. doi:10.1016/0021-9045(82)90033-8

Korneichuk N.P., Babenko V.F., Ligun A.A. Inequalities for derivatives and their applications, Nova Science Publishers, 1996.

Kofanov V.A. "Inequalities of various metrics for differentiable periodic functions", Ukrainian Math. J., 2015; 67(2): pp. 202-212. (in Russian) doi:10.1007/s11253-015-1076-2