It is well known that the system of translates $\{T_k\phi:k\in\mathbb{Z}\}$ is a
*Riesz* sequence in $L^2(\mathbb{R})$ if and only if there exist $A,B>0$ such that
\begin{equation}
A\leq\sum_{k\in\mathbb{Z}}|\widehat{\phi}(\xi+k)|^2\leq B\hspace{.5 cm}a.e.\ \xi\in[0,1],
\end{equation}
where $\widehat{\phi}$ denotes the *Fourier* transform of $\phi$. This result is very
important in time-frequency analysis especially in constructing wavelet basis for
$L^2(\mathbb{R})$ using *multiresolution analysis* technique and also in studying sampling
problems in a shift-invariant space.

In this talk, we ask a similar question for the system of left translates
$\{ L_\gamma\phi:\gamma\in\Gamma\}$ on the *Heisenberg* group $\mathbb{H}^n$, where
$\phi\in L^2(\mathbb{H}^n)$ and $\Gamma$ is a lattice in $\mathbb{H}^n$. We take $\Gamma=
\{(2k,l,m):k,l\in\mathbb{Z}^n,m\in\mathbb{Z}\}$ as the standard lattice in order
to avoid computational complexity. Recently it has been proved that if
$\phi\in L^2(\mathbb{H}^n)$ is such that
\begin{equation}
\sum_{r\in\mathbb{Z}}\left\langle \widehat{\phi}(\lambda+r),\widehat{L_{(2k,l,0)}\phi}(\lambda+r)
\right\rangle_{\mathcal{B}_2}|\lambda+r|^n=0\ a.e.\ \lambda\in(0,1],
\end{equation}
for all $(k,l)\in\mathbb{Z}^{2n}\setminus\{(0,0)\}$, then $\{L_{(2k,l,m)}\phi:k,l\in\mathbb{Z}^n,
m\in\mathbb{Z}\}$ is a *Riesz* sequence if and only if there exist $A,B>0$ such that
\begin{equation}
A\leq \sum_{r\in\mathbb{Z}}\left|\widehat{\phi}(\lambda+r)\right|_{\mathcal{B}_2}^2|\lambda+r|^n\leq
B\ \ a.e.\ \lambda\in(0,1].
\end{equation}
Here $\widehat{\phi}$ denotes the group *Fourier* transform of $\phi$ and $\mathcal{B}_2$ denotes the
*Hilbert* space of *Hilbert-Schmidt* operators on $L^2(\mathbb{R}^n)$. In the absence of the above condition,
the requirement of *Riesz* sequence is given in terms of the Gramian of the system
$\{\tau\left(L_{(2k,l,0)}\phi\right)(\lambda):k,l\in\mathbb{Z}^n\}$ for $\lambda\in(0,1]$, where $\tau$
is the fiber map. We shall discuss these results in the talk along with the computational issues.

- All seminars.
- Seminars for 2021

Last updated: 11 Oct 2024