Multi-channel sampling on shift-invariant spaces with frame generators (Q2893481)

Shift-invariant spaces have gained increasing recognition during the last 15 years due in major part to their connection with sampling theory. This paper considers functions \(f\) in a shift-invariant space \(V(\phi)= \overline{\text{span}}_{L^2} \{\phi(t-n): n \in \mathbb Z\}\) where \(\phi\) is a continuous Riesz generator and the spectrum of \(V(\phi)\) is multi-banded. Suppose that \(N\) linear time-invariant systems \({\mathcal L_j}\), \(1 \leq j \leq N\), are defined on \(V(\phi)\). The paper derives a formula such that, given \(r, N\) positive integers and real numbers \(0 \leq a_j<r\), for \(1\leq j \leq N\), any \(f \in V(\phi)\) can be expressed as NEWLINE\[NEWLINE f(t) = \sum_{j=1}^N \sum_{n \in {\mathbb Z}} ({\mathcal L_j}f)(a_j+rn) \, S_{j,n}(t) ,NEWLINE\]NEWLINE where \(S_{j,n}(t)\) is a sequence of sampling functions forming a frame or a Riesz basis in \(V(\phi)\). When \(N=r=1\) and the linear time-invariant system \({\mathcal L}\) is the identity operator, the above formula reduces to the classical sampling formula in \(V(\phi)\): NEWLINE\[NEWLINE f(t) = \sum_{n \in {\mathbb Z}} f(a+ n) \, S_{n}(t).NEWLINE\]