Skip to main content
SpringerLink
Log in

Ultimate generalization to monotonicity for uniform convergence of trigonometric series

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n→∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series Σ ∞n=1 a n sin nx is lim n→∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Askey R. Smoothness conditions for Fourier series with monotone coefficients. Acta Sci Math (Szeged), 1967, 28: 169–171

    MATH  MathSciNet  Google Scholar 

  2. Belov A S. On sequential estimate of best approximations and moduli of continuity by sums of trigonometric series with quasimonotone coefficients (in Russian). Matem Zametki, 1992, 51: 132–134

    MathSciNet  Google Scholar 

  3. Bray W O, Stanojevic V B. On the integrability of complex trigonometric series. Proc Amer Math Soc, 1985, 51: 51–58

    MathSciNet  Google Scholar 

  4. Chaundy T W, Jolliffe A E. The uniform convergence of a certain class of trigonometric series. Proc London Math Soc, 1916, 15: 214–216

    Article  Google Scholar 

  5. Konovich T O. Estimate for the best approximation of periodic functions in the metric L p (in Ukrainian). In: The Extremal Problems in the Theory of Functions and Related Problems. Kiev Institute of Mathematics, Ukrainian Academy of Science, 2003, 83–88

  6. Konyushkov A A. Best approximation by trigonometric polynomials and Fourier coefficients (in Russian). Math Sbornik, 1958, 44: 53–84

    Google Scholar 

  7. Le R J, Zhou S P. A new condition for the uniform convergence of certain trigonometric series. Acta Math Hungar, 2005, 108: 161–169

    Article  MATH  MathSciNet  Google Scholar 

  8. Leindler L. On the uniform convergence and boundedness of a certain class of sine series. Anal Math, 2001, 27: 279–285

    Article  MATH  MathSciNet  Google Scholar 

  9. Leindler L. A new class of numerical sequences and its applications to sine and cosine series. Anal Math, 2002, 28: 279–286

    Article  MATH  MathSciNet  Google Scholar 

  10. Leindler L. Best approximation and Fourier coefficients. Anal Math, 2005, 31: 117–129

    Article  MATH  MathSciNet  Google Scholar 

  11. Leindler L. On the relationships of seven numerical sequences. Acta Math Hungar, 2007, 114: 227–234

    Article  MATH  MathSciNet  Google Scholar 

  12. Mazhar S M. On strong approximation of continuous functions. Anal Math, 2003, 29: 281–287

    Article  MATH  MathSciNet  Google Scholar 

  13. Nurcombe J R. On the uniform convergence of sine series with quasimonotone coefficients. J Math Anal Appl, 1992, 166: 577–581

    Article  MATH  MathSciNet  Google Scholar 

  14. Shah S M. Trigonometric series with quasi-monotone coefficients. Proc Amer Math Soc, 1962, 13: 266–273

    MATH  MathSciNet  Google Scholar 

  15. Stanojevic V B. O-regularly varying convergence moduli of Fourier and Fourier-Stieltjes series. Math Ann, 1987, 279: 103–115

    Article  MATH  MathSciNet  Google Scholar 

  16. Stanojevic V B. L 1-convergence of Fourier series with O-regularly varying quasimonotone coefficients. J Approx Theory, 1990, 60: 168–173

    Article  MATH  MathSciNet  Google Scholar 

  17. Stanojevic V B. Fourier and trigonometric transforms with complex coefficients regularly varying in mean. Lect Notes Pure Appl Math, 1994, 157: 423–432

    MathSciNet  Google Scholar 

  18. Telyakovskii S A. On the convergence in the metric of L of trigonometic series with rarely changing coefficients (in Russian). Trudy Mat Inst Steklov, 1991, 200: 322–326

    Google Scholar 

  19. Telyakovskii S A. Estimation of the moduli of continuity of one-variable functions in the metric of L in terms of Fourier coefficients. Ukrainian Math J, 1994, 46: 671–678

    Article  MathSciNet  Google Scholar 

  20. Telyakovskii S A. Uniform convergence of trigonometric series with rarely changing coefficients. Math Notes, 2001, 70: 553–559

    Article  MathSciNet  Google Scholar 

  21. Telyakovskii S A. Estimates for the integral modulus of continuity of functions with rarely changing Fourier coefficients (in Russian). Mat Sb, 2002, 193: 93–106

    Article  MathSciNet  Google Scholar 

  22. Tikhonov S. Strong approximation of Fourier series and embedding theorems. Anal Math, 2005, 31: 183–194

    Article  MATH  MathSciNet  Google Scholar 

  23. Xie T F, Zhou S P. The uniform convergence of certain trigonometric series. J Math Anal Appl, 1994, 181: 171–180

    Article  MATH  MathSciNet  Google Scholar 

  24. Xie T F, Zhou S P. L 1-approximation of Fourier series of complex valued functions. Proc Royal Soc Edinburgh Ser A, 1996, 126: 343–353

    MATH  MathSciNet  Google Scholar 

  25. Yu D S, Zhou S P. A generalization of monotonicity condition and applications. Acta Math Hungar, 2007, 115: 247–267

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    SongPing Zhou

  2. Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, B2G 2W5, Canada

    SongPing Zhou & Ping Zhou

  3. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China

    DanSheng Yu

Authors
  1. SongPing Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ping Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. DanSheng Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to SongPing Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, S., Zhou, P. & Yu, D. Ultimate generalization to monotonicity for uniform convergence of trigonometric series. Sci. China Math. 53, 1853–1862 (2010). https://doi.org/10.1007/s11425-010-3138-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-3138-0

Keywords

MSC(2000)

Search

Navigation