Abstract
In this paper, we prove the Hardy’s and Cowling–Price’s uncertainty principles for the generalized Fourier transform associated to a Cherednik type operator on the real line.
Similar content being viewed by others
References
Anker J.-PH., Ayadi F., Sifi M.: Opdam function: Product formula and convolution structure in dimention, Adv. Pure Appl. Math. 3, 11–44 (2012)
L. Bou Attour and K. Trimèche, Uncertainty principle and L p−L q-sufficient pairs on Chébli–Trimèche hypergroups. Integral Transform Spec. Funct. 16, (2005), 625–637.
Cherednik I.: A unification of Knizhnik-Zamolodchikov equations and Dunkl operators via affine Hecke algebras, Invent. Math. 106, 411–432 (1991)
M.G. Cowling and J.F. Price, Generalizations of Heisenberg inequality, Lecture Notes in Math., 992. Springer, Berlin (1983), 443–449.
Eguichi M., Koizumi S., Kumahara K.: An L p version of Hardy theorem for the motion group. J. Austral. Math. Soc. Serie A. 68, 55–67 (2000)
Fitouhi A.: Heat polynomials for a singular differential operator on (0, ∞). J. Constr. Approx. 5, 241–270 (1989)
Gallardo L., Trimèche K.: An L p version of Hardy’s theorem for the Dunkl transform. J. Austr. Math. Soc. 77, 371–386 (2004)
Gallardo L., Trimèche K.: Positivity of the Jacobi–Cherednik intertwining operator and its dual. Adv. Pure Appl. Math. 1, 163–194 (2010)
Hardy G.H.: A theorem concerning Fourier transform. J. London Math. Soc. 8, 227–231 (1933)
Heckmann G.J., Opdam E.M.: Root systems and hypergeometric functions I. Compositio Math. 64, 329–352 (1987)
Mejjaoli H.: Qualitative uncertainty principles for the Opdam-Cherednik transform. Integral Transforms and Special Functions 25, 528–546 (2014)
Mourou M.A.: Transmutation operators and Paley–Wiener associated with a Cherednik type operator on the real line. Anal. Appl. 8, 387–408 (2010)
Opdam E.M.: Harmonic analysis for certain representations of graded Hecke algebras. Acta. Math. 175, 75–121 (1995)
Ray S.K., Sarkar R.P.: Cowling–Price theorem and characterization of heat kernel on symmetric spaces. Proc. Indian Acad. Sci. (Math. Sci.) 114, 159–180 (2004)
Schapira B.: Contributions to the hypergeometric function theory of Heckman and Opdam: sharpe stimates, Schwartz spaces, heat kernel. Geom. Funct. Anal. 18, 222–250 (2008)
Trimèche K.: Positivity of the transmutation operators associated with a Cherednik type operator on the real line. Advances in Pure and Applied Mathematics. 3, 361–376 (2013)
Yakubovich S.B.: Uncertainty principles for the Kontorovich-Lebedev transform. Math. Model. Anal. 13, 289–302 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Khalifa Trimèche on the occasion of his promotion to Professor Emeritus.
Rights and permissions
About this article
Cite this article
Mejjaoli, H. Cowling–Price’s and Hardy’s uncertainty Principles for the generalized Fourier transform associated to a Cherednik type operator on the real line. Arch. Math. 104, 377–389 (2015). https://doi.org/10.1007/s00013-015-0742-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-015-0742-9