The {A}malgan {B}alian {L}ow Theorem and \\time-frequency shift invariance

CABRELLI, CARLOS A.; MOLTER, URSULA; PFANDER, GÖTZ

El Escorial

Congreso; 10th International Conference on Harmonic Analysis and Partial Differential Equations; 2016

Universidad Autonoma de Madrid

The {\em Balian-Low Theorem} expresses the fact that time-frequency concentration and non redundancy are essentially incompatible. Specifically, if $\varphi\in L^2(\R)$, $\Lambda \subset \R^2$ is a lattice and the system $(\varphi, \Lambda)= \{e^{2\pi i \eta x} \varphi(x - u): (u, \eta) \in \Lambda\}$ is a Riesz basis for $L^2(\R)$, then $\varphi$ satisfies\begin{align*} \Big(\int (x-a)^2 |\varphi (x)|^2 \, dx \Big)\cdot \Big(\int (\omega-b)^2 |\widehat \varphi (\omega)|^2 \, d\omega \Big) =\infty, \quad a,b\in\R.\end{align*}The {\em Amalgam Balian-Low Theorem} states that if $(\varphi,\alpha\Z\times\beta\Z)$ is a Riesz basis for $L^2(\R)$, then $\varphi$ cannot belong to the Feichtinger algebra $S_0(\R)$, a class of functions decaying well in time and frequency. Precisely, \begin{equation*}S_0(\R) =\left\{f\in L^2(\R): V f(t,\nu)= \int f(x) e^{-(x-t)^2}e^{2\pi i x \nu}\, dx \in L^1(t,\nu)\right\}.\end{equation*}Note that $Vf(t,\nu) \in L^2(t,\nu) \cap L^\infty (t,\nu)$ for all $f\in L^2(\R)$ and the requirement $Vf(t,\nu) \in L^1(t,\nu)$ essentially necessitates $L^1$ decay of $f$ and of its Fourier transform $\widehat f$. This space is called the {\em Feichtinger algebra}.Let $T_u f(x)=f(x-u)$, and $M_\eta f(x)=e^{2\pi i \eta x} f(x)$, denote the usual translation and modulation operators, and let $\pi(u,\eta)=M_\eta T_u$, (with $u\in\R$ and $\eta\in\widehat \R$ the dual group of $\R$) denote the time-frequency shift. For $\varphi\in L^2(\R)$ and a lattice $\Lambda \subset \R\times \widehat \R$, let $\G(\varphi,\Lambda)$ denote the {\em Gabor spaces}, $\G(\varphi,\Lambda):=\overline{\spa\{\pi(\lambda)\varphi\}}$, where $\overline V$ is the closure of $V$ in $L^2(\R)$. In this talk we address the question whether there may exist a $\mu\in \R\times \widehat \R\setminus \Lambda$ with $\pi(\mu)\varphi \in \G(\varphi,\Lambda)$. The result relates the existence of such $\mu$, to the fact that $\varphi$ belongs (or does not belong) to the {\em smoothness space} $S_0(\R)$.We have\begin{theorem*} If $(\varphi,\Lambda)$ is a Riesz basis for its closed linear span $\mathcal G(\varphi,\Lambda)$ with $\varphi\in S_0(\R)$ and the density of the lattice $\Lambda$ is rational, then for any $(u,\eta)\notin \Lambda$ $\pi(u,\eta)\varphi \notin \mathcal G(\varphi,\Lambda)$ .% %%In the case $\Lambda=\alpha\Z\times\beta\Z$, then the condition $\varphi\in S_0(\R)$ can be replaced with the weaker condition that $Z_\alpha\varphi(x,\omega)=\sum_{n\in\Z} f(x+n\alpha)e^{-2\pi i \omega n\alpha}$ is continuous on $\R\times\widehat\R$. \end{theorem*}Note that $(\varphi,\Lambda)$ being a Riesz basis for $L^2(\R)$ implies that the density of $\Lambda$ equals $1 \in \Q$; and $\mathcal G(\varphi,\Lambda)=L^2(\R)$ implies that $\pi(u,\eta)\varphi\in \mathcal G(\varphi,\Lambda)$ for all $(u,\eta)\in\R \times \widehat \R$. Therefore the theorem implies that $\varphi\notin S_0(\R)$.