Heisenberg-type inequalities for the Weinstein operator

Abstract

The aim of this paper is to prove new uncertainty principles for the Weinstein operator. The first of these results is a sharp Heisenberg-type inequality for the Weinstein transform, that is, for $s\ge 1$ and $f\in L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)$, ${\Vert |x{|}^{s}f\Vert }_{L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)}{\Vert |\xi {|}^s{\mathcal{F}}_{\mathrm{W}}\left(f\right)\Vert }_{L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)}\ge {\left(\alpha +\frac{(d+1)}{2}\right)}^s\parallel f{\parallel }_{L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)}^2.$ The second result states that the previous inequality can be refined for an orthonormal basis for $L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)$, that is, if $\{{\phi }_n{\}}_{n=1}^{\mathrm{\infty }}$ is an orthonormal basis for $L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)$, then $\underset{n}{sup}(\parallel |x{|}^s{\phi }_n{\parallel }_{L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)}\parallel |\xi {|}^s{\mathcal{F}}_{\mathrm{W}}({\phi }_n\left){\parallel }_{L_{\alpha }^2\left({\mathbb{R}}_{+}^d\right)}\right)=\mathrm{\infty }.$ As a side result, we prove a new version of Heisenberg's uncertainty inequality for the Weinstein–Gabor transform, which states that the Weinstein–Gabor transform of a nonzero function with respect to a nonzero window function cannot be time and frequency concentrated around zero.

Keywords:

• Weinstein's operator
• Heisenberg's uncertainty inequality
• time–frequency concentration

1991 Mathematics Subject Classification:

• 42B10
• 44A20

Acknowledgments

The authors thank the anonymous referee for his/her careful reading of the manuscript that leads to a refinement presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).