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Almost everywhere convergence of inverse Fourier transforms


Authors: Leonardo Colzani, Christopher Meaney and Elena Prestini
Journal: Proc. Amer. Math. Soc. 134 (2006), 1651-1660
MSC (2000): Primary 42B10, 43A50; Secondary 42C15
DOI: https://doi.org/10.1090/S0002-9939-05-08329-2
Published electronically: October 18, 2005
MathSciNet review: 2204276
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Abstract: We show that if $ \log(2-\Delta)f\in L^2({\mathbb{R}}^d)$, then the inverse Fourier transform of $ f$ converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when $ \sqrt{\log(2-\Delta)}f\in L^2({\mathbb{R}}^d)$ and $ \log\log(4-\Delta)f\in L^2({\mathbb{R}}^d)$. We also consider sequential convergence for general elements of $ L^2({\mathbb{R}}^d)$.


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  • 1. G. Alexits, Convergence problems of orthogonal series, Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York, 1961. MR 0218827 (36:1911)
  • 2. Anthony Carbery, Dirk Gorges, Gianfranco Marletta, and Christoph Thiele, Convergence almost everywhere of certain partial sums of Fourier integrals, Bull. Lond. Math. Soc. 35 (2003), no. 2, 225-228. MR 1952399 (2003k:42027)
  • 3. Anthony Carbery, José L. Rubio de Francia, and Luis Vega, Almost everywhere summability of Fourier integrals, J. London Math. Soc. (2) 38 (1988), no. 3, 513-524. MR 0972135 (90e:42033)
  • 4. Anthony Carbery and Fernando Soria, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an $ L\sp 2$-localisation principle, Rev. Mat. Iberoamericana 4 (1988), no. 2, 319-337. MR 1028744 (91d:42015)
  • 5. Michael Christ, Javier Duoandikoetxea, and José L. Rubio de Francia, Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations, Duke Math. J. 53 (1986), no. 1, 189-209. MR 0835805 (88d:42032)
  • 6. Dirk Gorges, Convergence almost everywhere of partial sums of Fourier-integrals, Ph.D. thesis, Trier: Univ. Trier, Fachbereich IV, 50 p., 1998. MR 1952399 (2003k:42027)
  • 7. Blin Mao, Heping Liu, and Shanzhen Lu, Norm inequality with power weights for a class of maximal spherical summation operators, J. Beijing Norm. Univ., Nat. Sci. 1989 (1989), no. 2, 1-4 (Chinese. English summary). MR 1018436 (90h:42022)
  • 8. Christopher Meaney, On almost-everywhere convergent eigenfunction expansions of the Laplace-Beltrami operator, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 1, 129-131. MR 0662968 (83k:58093)
  • 9. Emmanuel Montini, On the capacity of sets of divergence associated with the spherical partial integral operator, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1415-1441 (electronic). MR 1946398 (2003h:42020)
  • 10. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • 11. A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21:6498)

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Additional Information

Leonardo Colzani
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Edificio U5, via Cozzi 53, 20125 Milano, Italy
Email: leonardo@matapp.unimib.it

Christopher Meaney
Affiliation: Department of Mathematics, Macquarie University, North Ryde NSW 2109, Australia

Elena Prestini
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy

DOI: https://doi.org/10.1090/S0002-9939-05-08329-2
Keywords: Rademacher-Menshov Theorem, inverse Fourier transform, series of orthogonal functions
Received by editor(s): December 27, 2004
Published electronically: October 18, 2005
Additional Notes: The second author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”. We are grateful to Fulvio Ricci and the Centro di Ricerca Matematica Ennio De Giorgi for their hospitality
The third author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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