Time frequency shift invariance and the Amalgam Balian Low Theorem

CARLOS CABRELLI; MOLTER, URSULA; PFANDER, GÖTZ

Rio de Janeiro

Congreso; Coloquio Brazileiro de Matematica; 2015

Sociedad Brasilera de matematica

The Balian-Low Theorem, a key result in time-frequency analysis, expresses the fact that time-frequency concentration and non redundancy are essentially incompatible. Specifically, if $\varphi\in L^2(\R)$, $\Lambda \subset \R^d$ 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*}We define the unitary operators, translation $T_u:L^2(\R)\longrightarrow L^2(\R)$, $T_u f(x)=f(x-u)$, modulation $M_\eta:L^2(\R)\longrightarrow L^2(\R)$, $M_\eta f(x)=e^{2\pi i \eta x} f(x)$, and time-frequency shift $\pi(u,\eta)=M_\eta T_u$, where $u\in\R$ and $\eta\in\widehat \R$, the dual group of $\R$ which is isomorphic to $\R$. For $\varphi\in L^2(\R)$ and a lattice $\Lambda= R\Z^2\subset \R\times \widehat \R$, $R\in\R^{2\times 2}$, {\em Gabor systems} as $(\varphi,\Lambda)= \{\pi(\lambda)\varphi\}_{\lambda\in\Lambda}$ and {\em Gabor spaces} as $\G(\varphi,\Lambda)=\overline{\spa\{\pi(\lambda)\varphi\}}$, where $\overline V$ is the closure of $V$ in $L^2(\R)$. This talk addresses the question whether there may exist a $\mu\in \R\times \widehat \R\setminus \Lambda$ with $\pi(\mu)\varphi \in \G(\varphi,\Lambda)$. To state our result, we recall the definition of the {\em smoothness space} $S_0(\R)$\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}.We can show that\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 $\pi(u,\eta)\varphi \notin \mathcal G(\varphi,\Lambda)$ for all $(u,\eta)\notin \Lambda$. 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*}This theorem generalizes the Amalgam Balian Low Theorem stated above. Indeed, $(\varphi,\Lambda)$ being a Riesz basis for $L^2(\R)$ implies that the density of $\Lambda$ equals $1$, that is, $(\alpha\beta)^{-1}=1\in\Q$ in case $\Lambda=\alpha\Z\times\beta\Z$, and $\mathcal G(\varphi,\Lambda)=L^2(\R)$ implies that $\pi(u,\eta)\varphi\in \mathcal G(\varphi,\Lambda)$ for all $(u,\eta)\in\R \times \widehat \R$, so the theorem implies that $\varphi\notin S_0(\R)$.