Skip to main content
SpringerLink
Log in

Uniform convergence of double sine transforms of general monotone functions

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We consider different classes of functions of two variables that satisfy general monotonicity conditions, and obtain necessary and sufficient conditions for the uniform convergence in the regular sense and in the sense of Pringsheim of double sine transforms of functions of such classes.

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. E. Berkson and T. A. Gillespie, Absolutely continuous functions of two variables and well-bounded operators, J. London Math. Soc. (2), 30 (1984), 305–321.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Booton, General monotone functions and their Fourier coefficients, J. Math. Anal. Appl., 426 (2015), 805–823.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. W. Chaundy and A. E. Jolliffe, The uniform convergence of a certain class of trigonometrical series, Proc. London Math. Soc., S2-15 (1916), 214–216.

    Article  MathSciNet  Google Scholar 

  4. J. A. Clarkson, On double Riemann-Stieltjes integrals, Bull. Amer. Math. Soc., 39 (1933), 929–936.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. A. Clarkson and C. R. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc., 35 (1933), 824–854.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Debernardi, Uniform convergence of sine integrals with general monotone functions, Math. Nachr., to appear.

  7. M. Dyachenko, E. Liflyand and S. Tikhonov, Uniform convergence and integrability of Fourier integrals, J. Math. Anal. Appl., 372 (2010), 328–338.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Dyachenko, A. Mukanov and S. Tikhonov, Uniform convergence of trigonometric series with general monotone coefficients (preprint).

  9. M. Dyachenko and S. Tikhonov, Integrability and continuity of functions represented by trigonometric series: coefficients criteria, Studia Math., 193 (2009), 285–306.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Feng, V. Totik and S. P. Zhou, Trigonometric series with a generalized monotonicity condition, Acta Math. Sin. (Engl. Ser.), 30 (2014), 1289–1296.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. M. Grigoriev, Y. Sagher and T. R. Savage, General monotonicity and interpolation of operators, J. Math. Anal. Appl., 435 (2016), 1296–1320.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. H. Hardy, On the convergence of certain multiple series, Proc. London Math. Soc., S2-1 (1903), 124–128.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series, Vol. I, Dover Publications Inc. (New York, 1927).

    MATH  Google Scholar 

  14. P. Kórus, Remarks on the uniform and L 1-convergence of trigonometric series, Acta Math. Hungar., 128 (2010), 369–380.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Kórus and F. Móricz, Addendum to the paper “Generalizations to monotonicity for uniform convergence of double sine integrals over \(\bar {\mathbb{R}}_ + ^2\)” (Studia Math. 201 (2010), 287–304), Studia Math., 212 (2012), 285–286.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Kórus and F. Móricz, Generalizations to monotonicity for uniform convergence of double sine integrals over \(\bar {\mathbb{R}}_ + ^2\), Studia Math., 201 (2010), 287–304.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Leindler, On the uniform convergence of single and double sine series, Acta Math. Hungar., 140 (2013), 232–242.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Liflyand and S. Tikhonov, A concept of general monotonicity and applications, Math. Nachr., 284 (2011), 1083–1098.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Liflyand and S. Tikhonov, Extended solution of Boas’ conjecture on Fourier transforms, C. R. Math. Acad. Sci. Paris, 346 (2008), 1137–1142.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Liflyand and S. Tikhonov, The Fourier transforms of general monotone functions, in: Analysis and mathematical physics, Trends in Mathematics, Birkhäuser (Basel, 2009), pp. 377–395.

    Chapter  Google Scholar 

  22. F. Móricz, On the convergence of double integrals and a generalized version of Fubini’s theorem on successive integration, Acta Sci. Math. (Szeged), 78 (2012), 469–487.

    MathSciNet  MATH  Google Scholar 

  23. F. Móricz, On the convergence in a restricted sense of multiple series, Anal. Math., 5 (1979), 135–147.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Móricz, On the uniform convergence of double sine integrals over \(\bar {\mathbb{R}}_ + ^2\), Analysis (Munich), 31 (2011), 191–204.

    MathSciNet  Google Scholar 

  25. F. Móricz, On the uniform convergence of sine integrals, J. Math. Anal. Appl., 354 (2009), 213–219.

    Article  MathSciNet  MATH  Google Scholar 

  26. F. Móricz, Pointwise convergence of double Fourier integrals of functions of bounded variation over R2, J. Math. Anal. Appl., 424 (2005), 1530–1543.

    Article  MATH  Google Scholar 

  27. F. Riesz and B. Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, translated from the second French edition by Leo F. Boron, Dover Publications Inc. (New York, 1990).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326 (2007), 721–735.

    Article  MathSciNet  MATH  Google Scholar 

  30. W. H. Young, On Multiple Fourier Series, Proc. London Math. Soc., S2-11 (1912), 133–184.

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Zygmund, Trigonometric series, Vols. I, II, third edition with a foreword by Robert A. Fefferman, Cambridge University Press (Cambridge, 2002).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Recerca Matemàtica and Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    A. Debernardi

Authors
  1. A. Debernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Debernardi.

Additional information

This research was partially supported by the CERCA Programme of the Generalitat de Catalunya, Centre de Recerca Matemàtica in Barcelona and the MTM2014–59174–P grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Debernardi, A. Uniform convergence of double sine transforms of general monotone functions. Anal Math 43, 193–217 (2017). https://doi.org/10.1007/s10476-017-0304-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-017-0304-4

Key words and phrases

Mathematics Subject Classification

Search

Navigation