Uncertainty Principles for the Jacobi Transform

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 $f$.