Asymmetric multi-channel sampling in shift invariant spaces
Let \(V(\phi)\) be the principal shift invariant subspace consisting of the closure in \(L^2(\mathbb{R})\) of the finite linear combinations of the integer shifts \(\phi(\cdot-k)\) (\(k\in\mathbb{Z}\)) of \(\phi\). Here it is assumed these shifts form a Riesz basis for \(V(\phi)\). For \(1\leq j\leq N\), let \(L_j(t)\) be a linear, time-invariant (LTI) system with impulse response \(\ell_j(t)\). One seeks a frame or Riesz basis \(\{s_{j,k}\}\) (\(1\leq j\leq N\), \(k\in\mathbb{Z}\)) such that one has the generalized reconstruction from samples \[ f(t) =\sum_{j=1}^N \sum_{k=-\infty}^\infty L_j (f) (\sigma_j+r_j k)\, s_{j,k}(t) \] with convergence in \(L^2(\mathbb{R})\) whenever \(f\in V(\phi)\). Three types of LTIs are considered: the impulses are either shifted Diracs, or are \(L^2\) functions, or have Fourier transforms in \(L^2\cup L^\infty\) when \(H_\phi(\xi)=\sum |\widehat{\phi}(\xi+k)|^2\in L^2[0,2\pi]\). Here, \(\widehat{\phi}(\xi)=\int \phi(t)\, e^{-it\xi}\, dt\). Let \(\alpha_G={\text{ess\, inf}}\{\lambda_m(\xi)\}\) and \(\beta_G={\text{ess\, sup}}\{\lambda_M(\xi)\}\), the respective essential infima and suprema of \(\lambda_m\) and \(\lambda_M\), the smallest and largest eigenvalues of the \(r\times r\) matrix \(G^\ast(\xi) G(\xi)\) where \(G(\xi)\) is the transpose of the matrix \[ [\mathcal{D} g_{1,1}(\xi),\dots, \mathcal{D} g_{1,r/r_1}(\xi),\dots, \mathcal{D} g_{N,1}(\xi),\dots, \mathcal{D} g_{N,r/r_N}(\xi)] \] with \(g_j(\xi)=Z\psi_j (\sigma_j,\xi)/2\pi\) (\(\psi_j=L_j(\phi)\)) and \(g_{j,n}=g_j(\xi) e^{ir_j (n-1)\xi}\) where \(r=\{r_1,\dots, r_N\}\) and \(Z\) denote the Zak transform. The main result (Theorem 3.4) states that there is a Riesz basis \(\{s_{j,m_j,k}(t):\, 1\leq j\leq N, 1\leq m_j\leq r/r_j,k\in\mathbb{Z}\}\) of \(V(\phi)\) such that \[ f(t)=\sum_{j=1}^N \sum_{m_j=1}^{r/r_j} \sum_{k=-\infty}^\infty L_j (f) (\sigma_j +r_j (m_j-1)+rk)\, s_{j,m_j,k} (t) \] whenever \(f\in V(\phi)\), if and only if \(0<\alpha_G<\beta_G<\infty\) and \(\sum_{j=1}^N 1/r_j=1\). In this case, \(s_{j,m_j,k}(t)=s_{j,m_j}(t-k)\) and \(L_j(s_{q,m_q})(\sigma_j+r_j (m_j-1)+rk)=\delta_{j,q}\delta_{k,0}\) for \(1\leq j,q\leq N\) and \(k\in \mathbb{Z}\). The interpolating functions \(s_{j,m_j,k}\) are worked out in the proof. The condition \(\sum 1/r_j=1\) says that, in essence, the {\textit{average}} sampling rate is unit rate.
scientific article; zbMATH DE number 5687676