Skip to main content
SpringerLink
Log in

Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

It is proved that a trigonometric cosine series of the form Σ Emphasis>=0/∞ n a n cos(nx) with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.

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. P. Turán, “Egy Steinhausfele problémárol,”Mat. Lapok.,4, 263–275 (1953).

    MathSciNet  Google Scholar 

  2. H. Helson, “Proof of a conjecture of Steinhaus,”Proc. Nat. Acad. Sci. USA.,40, 205–206 (1954).

    MATH  MathSciNet  Google Scholar 

  3. N. K. Bari,Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  4. M. Weiss, “On a problem of J. E. Littlewood,”J. London Math. Soc.,34, 217–221 (1959).

    MATH  MathSciNet  Google Scholar 

  5. Y. Katsnel'son, “Trigonometric series with positive partial sums,”Bull. Amer. Math. Soc.,71, 718–719 (1965).

    MathSciNet  Google Scholar 

  6. R. E. Edwards,Fourier Series. A Modern Introduction, Vol. 2, New York (1979).

  7. A. Zygmund,Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge (1959).

    Google Scholar 

  8. A. S. Belov, “Coefficients of trigonometric cosine series with nonnegative partial sums,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],190, 3–21 (1989).

    MATH  MathSciNet  Google Scholar 

  9. A. S. Belov, “Expressions for the upper bounds of the partial sums of trigonometric series in terms of their lower bounds,”Mat. Sb. [Math. USSR-Sb.],183, No. 11, 55–74 (1992).

    MATH  MathSciNet  Google Scholar 

  10. A. S. Belov, “Coefficients of trigonometric series with nonnegative partial sums,”Mat. Zametki [Math. Notes],41, No. 2, 152–158 (1987).

    MATH  MathSciNet  Google Scholar 

  11. W. Rudin, “Some theorems on Fourier coefficients,”Proc. Amer. Math. Soc.,10, No. 4, 855–859 (1959).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ivanovo State University, USSR

    A. S. Belov

Authors
  1. A. S. Belov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 24–41, January, 1996.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belov, A.S. Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums. Math Notes 59, 18–30 (1996). https://doi.org/10.1007/BF02312461

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02312461

Keywords

Search

Navigation