Hiroshima Mathematical Journal

Uncertainty principles for the Dunkl transform

Takeshi Kawazoe and Hatem Mejjaoli

Full-text: Open access

Abstract

The Dunkl transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price’s theorem, Beurling’s theorem and Donoho-Stark’s uncertainty principle are obtained for the Dunkl transform.

Article information

Source
Hiroshima Math. J., Volume 40, Number 2 (2010), 241-268.

Dates
First available in Project Euclid: 2 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1280754424

Digital Object Identifier
doi:10.32917/hmj/1280754424

Mathematical Reviews number (MathSciNet)
MR2680659

Zentralblatt MATH identifier
1214.43008

Subjects
Primary: 35C80
Secondary: 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] 43A32: Other transforms and operators of Fourier type

Keywords
Dunkl transform Hardy’s type theorem Cowling-Price’s theorem Beurling’s theorem Miyachi’s theorem Donoho-Stark’s uncertainty principle

Citation

Kawazoe, Takeshi; Mejjaoli, Hatem. Uncertainty principles for the Dunkl transform. Hiroshima Math. J. 40 (2010), no. 2, 241--268. doi:10.32917/hmj/1280754424. https://projecteuclid.org/euclid.hmj/1280754424


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References

  • M. Benedicks, On Fourier transforms of function supported on sets of finite Lebesgue measure, J. Math. Anal. Appl., 106 (1985), 180-183.
  • A. Beurling, The collect works of Arne Beurling, Birkhäuser. Boston, 1989, 1-2.
  • A. Bonami, B. Demange and P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoamericana, 19 (2002), 22-35.
  • F. Chouchene, R. Daher, T. Kawazoe and H. Mejjaoli, Miyachi's theorem for the Dunkl transform on $\R^d$, Preprint 2007.
  • R. Daher, T. Kawazoe and H. Mejjaoli, A generalization of Miyachi's theorem, J. Math. Soc. Japan. Vol. 61 N.2 (2009), 551-558.
  • M.G. Cowling and J.F. Price, Generalizations of Heisenberg inequality, Lecture Notes in Math., 992. Springer, Berlin (1983), 443-449.
  • D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49 (1989), 906-931.
  • C.F. Dunkl, Differential-difference operators associated to reflection group, Trans. Am. Math. Soc., 311 (1989), 167-183.
  • C.F. Dunkl, Integral kernels with reflection group invariant, Can. J. Math., 43 (1991), 1213-1227.
  • C.F. Dunkl, Hankel transforms associated to finite reflection groups, Contemp. Math., 138 (1992), 123-138.
  • M.F.E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993), 147-162.
  • L. Gallardo and K. Trimèche, An $L^p$ version of Hardy's theorem for the Dunkl transform, J. Aust. Math. Soc., 77 (2004), 371-385.
  • G.H. Hardy, A theorem concerning Fourier transform, J. London Math. Soc., 8 (1933), 227-231.
  • L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. För Math., 2 (1991), 237-240.
  • H.J. Landau and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell. Syst. Tech. J., 40 (1961), 65-84.
  • H. Mejjaoli and K. Trimèche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integ. Transf. and Special Funct., 12 (2001), 279-302.
  • H. Mejjaoli and K. Trimèche, Hypoellipticity and hypoanaliticity of the Dunkl Laplacian operator, Integ. Transf. and Special Funct., 15 (2004), 523-548.
  • H. Mejjaoli, An analogue of Beurling-Hörmander's theorem associated with Dunkl-Bessel operator, Fract. Calc. Appl. Anal. 9 (2006), no. 3, 247-264.
  • H. Mejjaoli and K. Trimèche, A Variant of Cowling-Price's theorem for the Dunkl transform on $\R$, J. Math. Anal. Appl., 345 (2008), 593-606.
  • A. Miyachi, A generalization of theorem of Hardy, Harmonic Analysis Seminar held at Izunagaoka, Shizuoka-Ken, Japon 1997, 44-51.
  • G.W. Morgan, A note on Fourier transforms, J. London Math. Soc., 9 (1934), 188-192.
  • M. Rösler, Positivity of Dunkl's intertwining operator, Duke. Math. J., 98 (1999), 445-463.
  • M. Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc., 355 (2003), 2413-2438.
  • C. Pfannschmidt, A generalization of the theorem of Hardy: A most general version of the uncertainty principle for Fourier integrals, Math. Nachr., 182 (1996), 317-327.
  • S.K. Ray and R.P. Sarkar, Cowling-Price theorem and characterization of heat kernel on symmetric spaces, Proc. Indian Acad. Sci. (Math. Sci.), 114 (2004), 159-180.
  • S. Parui and R.P. Sarkar, Beurling's theorem and $L^p-L^q$ Morgan's theorem for step two nilpotent Lie groups, Preprint 2007.
  • D. Slepian and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell. Syst. Tech. J., 40 (1961), 43-63.
  • S. Thangavelu and Y. Xu, Convolution operator and maximal functions for Dunkl transform, J. d'Analyse Mathematique, 97 (2005), 25-56.
  • K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integ. Transf. and Special Funct., 12 (2001), 349-374.
  • K. Trimèche, \em Paley-Wiener theorems for Dunkl transform and Dunkl translation operators, Integ. Transf. and Special Funct., 13 (2002), 17-38.
  • K. Trimèche, An analogue of Beurling-Hörmander's theorem for the Dunkl transform, Global J. of Pure and Appl. Math., 5 (2007), 342-357.