Advertisement

Hankel Transforms of General Monotone Functions

  • Alberto Debernardi
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel–Olivier test for real-valued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.

Keywords

Hankel transform Boundedness Uniform convergence General monotonicity Cosine series 

AMS 2010 Subject Classification

Primary: 42A38; Secondary: 42A20 44A20 

Notes

Acknowledgement

This research was partially funded by the ERC starting grant No. 713927, the ISF grant No. 447/16, the CERCA Programme of the Generalitat de Catalunya, Centre de Recerca Matemàtica, and the MTM-2014-59174-P grant.

References

  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series (U.S. Government, Washington, DC, 1964)Google Scholar
  2. 2.
    T.W. Chaundy, A.E. Jolliffe, The uniform convergence of a certain class of trigonometrical series. Proc. Lond. Math. Soc. S2-15(1), 214–216 (1916)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Debernardi, Uniform convergence of Hankel transforms. J. Math. Anal. Appl. 468(2), 1179–1206 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Debernardi, Uniform convergence of sine integrals with general monotone functions. Math. Nachr. 290(17–18), 2815–2825 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Debernardi, E. Liflyand, S. Tikhonov, M. Zeltser, Pringsheim convergence and generalized monotonicity (Preprint)Google Scholar
  6. 6.
    M. Dyachenko, S. Tikhonov, Smoothness properties of functions with general monotone Fourier coefficients. J. Fourier Anal. Appl. 24(4), 1072–1097 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Dyachenko, E. Liflyand, S. Tikhonov, Uniform convergence and integrability of Fourier integrals. J. Math. Anal. Appl. 372, 328–338 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Dyachenko, A. Mukanov, S. Tikhonov, Uniform convergence of trigonometric series with general monotone coefficients (Preprint)Google Scholar
  9. 9.
    A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953)zbMATHGoogle Scholar
  10. 10.
    A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954)zbMATHGoogle Scholar
  11. 11.
    L. Feng, V. Totik, S.P. Zhou, Trigonometric series with a generalized monotonicity condition. Acta Math. Sin. Engl. Ser. 30(8), 1289–1296 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L.J. Landau, Bessel functions: monotonicity and bounds. J. Lond. Math. Soc. (2) 61, 197–215 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications. Math. Nachr. 284(8–9), 1083–1098 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. Liflyand, S. Tikhonov, M. Zeltser, Extending tests for convergence of number series. J. Math. Anal. Appl. 377, 194–206 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.Y. Olenko, Upper bound on \(\sqrt {x}J_\nu (x)\) and its applications. Integral Transforms Spec. Funct. 17(6), 455–467 (2006)Google Scholar
  16. 16.
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University, Princeton, NJ, 1971)zbMATHGoogle Scholar
  17. 17.
    S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326, 721–735 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Tikhonov, Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153, 19–39 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Zygmund, Trigonometric Series: Vol. I, II, 3rd edn. With a foreword by Robert A. Fefferman (Cambridge University Press, Cambridge, 2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alberto Debernardi
    • 1
    • 2
  1. 1.Bar-Ilan UniversityRamat-GanIsrael
  2. 2.Centre de Recerca MatemàticaBellaterraSpain

Personalised recommendations