Journal of the Mathematical Society of Japan

A generalization of Miyachi's theorem

Radouan DAHER, Takeshi KAWAZOE, and Hatem MEJJAOLI

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The classical Hardy theorem on R, which asserts f and the Fourier transform of f cannot both be very small, was generalized by Miyachi in terms of L1+L and log+-functions. In this paper we generalize Miyachi's theorem for Rd and then for other generalized Fourier transforms such as the Chébli-Trimèche and the Dunkl transforms.

Article information

J. Math. Soc. Japan, Volume 61, Number 2 (2009), 551-558.

First available in Project Euclid: 13 May 2009

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Zentralblatt MATH identifier

Primary: 43A85: Analysis on homogeneous spaces
Secondary: 43A62: Hypergroups 43A32: Other transforms and operators of Fourier type 44A12: Radon transform [See also 92C55] 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Hardy's theorem Miyachi's theorem Radon transform Dunkl transform Chébli-Trimèche transform


DAHER, Radouan; KAWAZOE, Takeshi; MEJJAOLI, Hatem. A generalization of Miyachi's theorem. J. Math. Soc. Japan 61 (2009), no. 2, 551--558. doi:10.2969/jmsj/06120551.

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  • W. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, Walter de Gruyter, Berlin, New-York, 1995.
  • A. Bonami, B. Demange and P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoamericana, 19 (2003), 23–55.
  • F. Chouchene, R. Daher, T. Kawazoe and H. Mejjaoli, Miyachi's theorem for the Dunkl transform, preprint, 2007.
  • F. Chouchene, M. Mili and K. Trimèche, Positivity of the intertwining operator and harmonic analysis associated with the Jacobi-Dunkl operator on $\mbi{R}$, J. Anal. Appl. (Singap.), 1 (2003), 387–412.
  • M. Cowling and J. F. Price, Generalizations of Heisenberg's inequality. Lecture Notes in Math., 992, Springer Verlag, 1983, pp. 443–449.
  • R. Daher and T. Kawazoe, Generalized Hardy's theorem for Jacobi transform, Hiroshima Math. J., 36 (2006), 331–337.
  • G.H. Hardy, A theorem concerning Fourier transforms, J. London. Math. Soc., 8 (1933), 227–231.
  • M. F. E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993), 147–162.
  • A. Miyachi, A generalization of theorem of Hardy, Harmonic Analysis Seminar held at Izunagaoka, Shizuoka-Ken, Japan 1997, pp. 44–51.
  • K. Trimèche, Generalized wavelets and hypergroups, Gordon and Breach Science, 1997.
  • K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms Spec. Funct., 12 (2001), 349–374.
  • K. Trimèche, Cowling-Price and Hardy theorems on Chébli-Trimèche hypergroups, Glob. J. Pure Appl. Math., 1 (2005), 286–305.