Tokyo Journal of Mathematics

Uncertainty Principles for the Jacobi Transform


Full-text: Open access


We obtain some uncertainty inequalities for the Jacobi transform $\hat f_{\alpha,\beta}(\lambda)$, where we suppose $\alpha, \beta\in\mathbb{R}$ and $\rho=\alpha+\beta+1\geq 0$. As in the Euclidean case, analogues of the local and global uncertainty principles hold for $\hat f_{\alpha,\beta}$. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When $\beta\leq 0$ or $\beta\leq\alpha$, the $L^2$-norm of $\hat f_{\alpha,\beta}(\lambda)\lambda$ is estimated below by the $L^2$-norm of $\rho f(x)(\cosh x)^{-1}$. Otherwise, a similar inequality holds. Especially, when $\beta>\alpha+1$, the discrete part of $f$ appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on $SU(1,1)$. Then the corresponding uncertainty principle depends, not uniformly on the $K$-types of $f$.

Article information

Tokyo J. Math., Volume 31, Number 1 (2008), 127-146.

First available in Project Euclid: 27 August 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


KAWAZOE, Takeshi. Uncertainty Principles for the Jacobi Transform. Tokyo J. Math. 31 (2008), no. 1, 127--146. doi:10.3836/tjm/1219844827.

Export citation


  • G. van Dijk and S. C. Hille, Canonical representations related to Hyperbolic spaces, J. Funct. Anal., 147, 1997, 109–139.
  • T. Kawazoe and J. Liu, On Hardy's theorem on $SU(1,1)$, preprint, 2005.
  • T. H. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat., 13, 1975, 145–159.
  • T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie Groups, Special functions, R. Askey et al. (eds.), D. Reidel Publishing Company, Dordrecht, 1984, 1–84.
  • J. F. Price and A. Sitaram, Local uncertainty inequalities for locally compact groups, Trans. Amer. Math. Soc., 308, 1988, 105–114.
  • P. Sally, Analytic Continuation of The Irreducible Unitary Representations of The Universal Covering Group of $SL(2,\Bbb R)$, Memoirs of the Amer. Math. Soc., 69, Amer. Math. Soc., Providence, Rhode Island, 1967.
  • A. Sitaram, M. Sundari and S. Thangavelu, Uncertainty principles on certain Lie groups, Proc. Indian Acad. Sci., 105, 1995, 135–151.
  • M. Sugiura, Unitary Representations and Harmonic Analysis, Second Edition, North-Holland, Amsterdam, 1990.
  • S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups, Progress in Mathematics, Birkhäuser, Boston, 2003.