Asymmetric multi-channel sampling in shift invariant spaces
type: Item
instance of: scholarly article
title: Asymmetric multi-channel sampling in shift invariant spaces
zbMATH Open document ID: 1257.94012
author: B. E. Eshmatov
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-03-29T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 94A20
zbMATH DE Number: 5687676
zbMATH Keywords: shift-invariant space
zbMATH Keywords: multi-channel sampling
zbMATH Keywords: frame
zbMATH Keywords: Riesz basis
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.12.005
OpenAlex ID: W2058604216
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DOI: 10.1016/J.JMAA.2009.12.005
description: scientific article; zbMATH DE number 5687676