Sampling and recovery of multidimensional bandlimited functions via frames (Q968826)

Let \(\mathcal F\) and \({\mathcal F}^{-1}\) denote the \(d\)-dimensional \(L_2\) direct and inverse Fourier transforms. The author proves the following multidimensional oversampling theorem for bandlimited functions. Let \(E\) such that \(0\in E\subset \mathbb{R}^d\) be compact and for all \(\lambda>1\), \(E\subset\text{int}(\lambda E)\). Choose \(\{t_n\}_{n\in{\mathbb N}}\subset\mathbb{R}^d\) such that \(\{f_n\}_{n\in{\mathbb N}}\), defined by \(f_n(\cdot)=e^{i(\cdot,t_n)}\), is a frame for \(L_2(E)\) with frame operator \(S\). Let \(\lambda_0>1\) with \({\mathcal F}^{-1}(g):\mathbb{R}^d\rightarrow\mathbb{R}\), \({\mathcal F}^{-1}(g)\in C^\infty\) where \({\mathcal F}^{-1}(g)|_{E}=1\) and \({\mathcal F}^{-1}(g)|_{(\lambda_0 E)^c}=0\). If \(\lambda\geq\lambda_0\) and \(f\in PW_E\), that is, bandlimited, then \(f(t)=(1/\lambda^d)\sum_{k\in{\mathbb N}}\left[\sum_{n\in{\mathbb N}} B_{kn}f(t_n/\lambda)\right]g(t-t_k/\lambda)\), \(t\in\mathbb{R}^d\), where \(B_{kn}=<S^{-1}f_n,S^{-1}f_k>_E\). Convergence of the sum is in \(L_2(\mathbb{R}^d)\), hence also uniform. Further, the map \(B:\ell_2({\mathbb N})\rightarrow\ell_2({\mathbb N})\) defined by \((y_k)_{k\in\mathbb N}\rightarrow (\sum_{n\in\mathbb N} B_{kn}y)_{k\in\mathbb N}\) is bounded linear, and is an onto isomorphism if and only if \(\{f_n\}_{n\in\mathbb N}\) is a Riesz basis for \(L_2(E)\). A computationally manageable simplification of this theorem is given. A generalization of Kadec's 1/4 theorem to higher dimension is considered.