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June 2008 Uncertainty Principles for the Jacobi Transform
Takeshi KAWAZOE
Tokyo J. Math. 31(1): 127-146 (June 2008). DOI: 10.3836/tjm/1219844827

Abstract

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 $f$.

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Takeshi KAWAZOE. "Uncertainty Principles for the Jacobi Transform." Tokyo J. Math. 31 (1) 127 - 146, June 2008. https://doi.org/10.3836/tjm/1219844827

Information

Published: June 2008
First available in Project Euclid: 27 August 2008

zbMATH: 1166.43004
MathSciNet: MR2426798
Digital Object Identifier: 10.3836/tjm/1219844827

Rights: Copyright © 2008 Publication Committee for the Tokyo Journal of Mathematics

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Vol.31 • No. 1 • June 2008
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