Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T21:08:33.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Convergence of Trigonometric Series with General Monotone Coefficients

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 January 2019

Mikhail Dyachenko ,
Askhat Mukanov  and
Sergey Tikhonov
Mikhail Dyachenko
Affiliation:
Department of Mechanics and Mathematics, Lomonosov Moscow State University, MSU, GSP-1, Moscow, 119991, Russia Email: dyach@mail.ru
Askhat Mukanov
Affiliation:
Centre de Recerca Matemàtica and Universitat Autónoma de Barcelona, Departament de Matematiques, Edifici C Facultat de Ciències, 08193 Bellaterra (Barcelona), Spain Kazakhstan Branch of Lomonosov Moscow State University, Kazhymukan St., 11, Astana, 010010, Kazakhstan Email: mukanov.askhat@gmail.com
Sergey Tikhonov
Affiliation:
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain, and Universitat Autónoma de Barcelona Email: stikhonov@crm.cat
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

Keywords

trigonometric seriesgeneral monotone sequenceuniform convergence

MSC classification

Primary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1445 - 1463
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This research was partially supported by RFFI no. 16-01-00350, MTM 2014-59174-P, 2014 SGR 289, the grants of Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (projects AP05131707, AP05133301), and by the CERCA Programme of the Generalitat de Catalunya.

References

Bernstein, S., Sur l’ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné . Mémoires de l’Académie Royale de Belgique (2) 4(1912), 1–104.Google Scholar
Boas , R. P. Jr., Integrability theorems for trigonometric transforms. Ergebnisse der Mathematik und ihrer Grenzgebiete, 38, Springer-Verlag, New York, 1967.Google Scholar
Chaundy, T. W. and Jolliffe, A. E., The uniform convergence of a certain class of trigonometrical series . Proc. London Math. Soc. (2) 15(1916), 214–216.Google Scholar
Dyachenko, M. and Tikhonov, S., Integrability and continuity of functions represented by trigonometric series: coefficients criteria . Studia Math. 193(2009), 285–306. https://doi.org/10.4064/sm193-3-5.Google Scholar
Dyachenko, M. and Tikhonov, S., General monotone sequences and convergence of trigonometric series. In: Topics in classical analysis and applications in honor of Daniel Waterman, World Sci., Hackensack, NJ, 2008, pp. 88–101.Google Scholar
Dyachenko, M. and Tikhonov, S., Smoothness properties of functions with general monotone Fouier coefficients . J. Fourier Anal. Appl.(2017). https://doi.org/10.1007/s00041-017-9553-7.Google Scholar
Fekete, M., Proof of three propositions of Paley . Bull. Amer. Math. Soc. 41(1935), 138–144. https://doi.org/10.1090/S0002-9904-1935-06036-7.Google Scholar
Feng, L., Totik, V., and Zhou, S. P., Trigonometric series with a generalized monotonicity condition . Acta Math. Sin. (Engl. Ser.) 30(2014), 8, 1289–1296. https://doi.org/10.1007/s10114-014-3496-6.Google Scholar
Flett, T. M., On the degree of approximation to a function by the Cesaro means of its Fourier series . Quart. J. Math. Oxford Ser. (2) 7(1956), 81–95. https://doi.org/10.1093/qmath/7.1.81.Google Scholar
Iosevich, A. and Liflyand, E., Decay of the Fourier transform. In: Analytic and geometric aspects. Birkhäuser/Springer, Basel, 2015.Google Scholar
Lebesgue, H., Sur la représentation trigonométrique approchée des fonctions satisfaisant á une condition de Lipschitz . Bull. Soc. Math. France 38(1910), 184–210. https://doi.org/10.24033/bsmf.859.Google Scholar
Leindler, L., On the uniform convergence and boundedness of a certain class of sine series . Anal. Math. 27(2001), 279–285. https://doi.org/10.1023/A:1014320328217.Google Scholar
Liflyand, E. and Tikhonov, S., A concept of general monotonicity and applications . Math. Nachr. 284(2011), 1083–1098. https://doi.org/10.1002/mana.200810262.Google Scholar
Rudin, W., Some theorems on Fourier coefficients . Proc. Amer. Math. Soc. 10(1959), 855–859. https://doi.org/10.1090/S0002-9939-1959-0116184-5.Google Scholar
Salem, R. and Zygmund, A., The approximation by partial sums of Fourier series . Trans. Amer. Math. Soc. 59(1946), 14–22. https://doi.org/10.1090/S0002-9947-1946-0015538-0.Google Scholar
Shapiro, H. S., Extremal problems for polynomials and power series. Thesis for S.M. Degree, Massachusetts Institute of Technology, 1951.Google Scholar
Tikhonov, S., Trigonometric series with general monotone coefficients . J. Math. Anal. Appl. 326(2007), 721–735. https://doi.org/10.1016/j.jmaa.2006.02.053.Google Scholar
Tikhonov, S., Best approximation and moduli smoothness: computation and equivalence theorems . J. Approx. Theory 153(2008), 19–39. https://doi.org/10.1016/j.jat.2007.05.006.Google Scholar
Zhou, S. P., Zhou, P., and Yu, D. S., Ultimate generalization to monotonicity for uniform convergence of trigonometric series . Sci. China Math. 53(2010), 1853–1862. https://doi.org/10.1007/s11425-010-3138-0.Google Scholar
Zygmund, A., Trigonometric series. Vol. I, II. Third ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002.Google Scholar