Abstract
The purpose of the present paper is to discuss the integrability of the Fourier transform of \(L^\infty \)-functions subject to a very weak decay condition. This will include the negative power of the logarithm. For a start, the case when \(d=1\) is investigated by the use of the second mean value theorem. Based on the discussion of this case, a passage to the weighted Hankel transform is done, which covers the integrability of the d-dimensional Fourier transform of the radial functions.
Similar content being viewed by others
References
Belinsky E., Trigub R.M.: A relation in the theory of Fourier series. Mat. Zametki 15 (1974), 679–682 (Russian)—English translation in Math. Notes 15 (1974), 405–407
Bochner, S.: Summation of multiple Fourier series by spherical means. Trans. Am. Math. Soc. 40, 175–207 (1936)
Colzani, L., Fontana, L., Laeng, E.: Asymptotic decay of Fourier, Laplace and other integral transforms. J. Math. Anal. Appl. 483(1), 123560 (2020)
De Carli, L., Gorbachev, D., Tikhonov, S.: Pitt inequalities and restriction theorems for the Fourier transform. Rev. Mat. Iberoam. 33(3), 789–808 (2017)
Estrada, R.: On radial functions and distributions and their Fourier transforms. J. Fourier Anal. Appl. 20, 301–320 (2014)
Gorbachev, D., Liflyand, E., Tikhonov, S.: Weighted norm inequalities for integral transforms. Indiana Univ. Math. J. 67(5), 1949–2003 (2018)
Grafakos, L.: Classical Fourier Analysis, Graduate texts in Mathematics, vol. 249. Springer, New York (2008)
Grafakos, L., Teschl, G.: On Fourier transforms of radial functions and distributions. J. Fourier Anal. Appl. 19, 167–179 (2013)
Iosevich, A., Liflyand, E.: Decay of the Fourier Transform: Analytic and Geometric Aspects. Birkhäuser, Basel (2014)
Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)
Liflyand, E.: Functions of Bounded Variation and their Fourier Transforms. Birkhäuser, Basel (2019)
Liflyand, E., Samko, S.: On Leray’s formula. In: Methods of Fourier Analysis and Approximation Theory, Appl. Numer. Harmon. Anal., pp. 139–146. Birkhauser/Springer, Cham (2016)
Liflyand, E., Tikhonov, S.: The Fourier transforms of general monotone functions. In: Analysis and Mathematical Physics, Trends in Mathematics, pp. 373–391. Birkhäuser, Basel (2009)
Liflyand, E., Trebels, W.: On asymptotics for a class of radial Fourier transforms. Z. Anal. Anwendungen 17, 103–114 (1998)
Nowak, A., Stempak, K.: A note on recent papers by Grafakos and Teschl, and Estrada. J. Fourier Anal. Appl. 20, 1141–1144 (2014)
Takagi, T.: Introduction to Analysis (in Japanese). Iwanami, Shoten (1938)
Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)
Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Appoximation of Functions. Kluwer, Dordrecht (2004)
Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (1959)
Acknowledgements
The authors are supported by Grant-in-Aid for Scientific Research (C), No. 19K03546, Japan Society for the Promotion of Science. The authors are thankful to anonymous referees for their kind comments which made the authors aware of many recent researches in this field.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Elijah Liflyand.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Okada, M., Sawano, Y. Weighted Hankel Transform and Its Applications to Fourier Transform. J Fourier Anal Appl 27, 23 (2021). https://doi.org/10.1007/s00041-021-09831-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-021-09831-4