Multi-channel sampling of band-limited functions and its well-posedness

Abstract

For any function of band-limited to $[-\pi, \pi]$, that is, $f \in PW_{\pi}$, if there exists $\alpha > 0$ such that $\alpha \leq |detH(\xi)|$, we have GSE; $\begin{displaymath} f(t)=\sum^{N}_{k=1} \sum_{n\in Z} g_{k}(nN) y_{k}(t-nN) \end{displaymath}$

where $H(\xi)$ is the transfer matrix of $N$ filters and

$\begin{displaymath} y_{k}(t) = \frac{1}{2\pi} \int^{\pi}_{-\pi} Y_{k} (\xi)e^{it\xi} d\xi \end{displaymath}

,which converges in $L^{2}(\mathbb{R})$ and uniformly on $\mathbb{R}$ and the GSE is well-posed. Moreover, $\{y_{k} (t-nN) | 1 \leq k \leq N, n \in \mathbb{Z}\}$ is Riesz basis of $PW_{\pi}$. And we find the determinant condition $\alpha \leq |detH(\xi)|$ is sufficient and necessary condition of being GSE Riesz basis expansion in $L^{2}(\mathbb{R})$.