Uncertainty principles for orthonormal sequences related to Laguerre hypergroup (Q2805340)

A time-frequency localization theorem for orthonormal sequences in \(L^2({\mathbb{R}}^d)\) that yields a number of uncertainty inequalities was obtained by \textit{E. Malinnikova} [J. Fourier Anal. Appl. 16, No. 6, 983--1006 (2010; Zbl 1210.42020)].NEWLINENEWLINEIn this paper, the author follows the same process in the case of the Fourier-Laguerre transform \({\mathcal{F}}_L\) on the Laguerre hypergroup \(\mathbb{K}=[0,+\infty)\times\mathbb{R}\). \({\mathcal{F}}_L\) gives an isometry between \(L^2(\mathbb{K}, dm_\alpha)\) and \(L^2(\mathbb{R}\times\mathbb{N},d\gamma_\alpha)\). Hence the time-frequency localization theorem for \({\mathcal{F}}_L\) is of the form: let \(\{\phi_n\}_{n=1}^N\) be an orthogonal system in \(L^2(\mathbb{K})\), \(T\) be a measurable subset of \(\mathbb{K}\) and \(W\) be a measurable set of \(\mathbb{R}\times\mathbb{N}\). Assume that NEWLINE\[NEWLINE \int_T|\phi_n|^2dm_\alpha=1-a_n^2 \text{ and } \int_W|{\mathcal{F}}_L\phi_n|^2d\gamma_\alpha=1-b_n^2. NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE \sum_{n=1}^N(1-\frac{3}{2}a_n-\frac{3}{2}b_n)\leq m_\alpha(T)\gamma_\alpha(W). NEWLINE\]NEWLINE According to Malinnikova's process, one can deduce other uncertainty inequalities from this inequality. However, since \(dm_\alpha\) and \(d\mu_\alpha\) are different, some calculation of the volume of balls and the constant \(K_\omega(\epsilon)\) in her paper have to be modified. These modifications are carried out explicitly.