The sampling theorem, \(L^ T_ q\)-approximation and \(\epsilon\)- dimension
For \(T=(T_ 1,\dots,T_ n)\in\mathbb{R}^ n\), \(T_ i>0\), \(i=1,\dots,n\), let \(Q_ T=\{x\in\mathbb{R}^ n\): \(x=(x_ 1,\dots,x_ n)\), \(| x_ i|\leq T_ i\), \(i=1,\dots,n\}\). A locally integrable function \(f\) on \(\mathbb{R}^ n\) is called bandlimited to a set \(G\subseteq\mathbb{R}^ n\) if the support of its distributional Fourier transform \(\hat f\) is contained in \(G\). The set of all functions in \(L^ p(G)\), \(1\leq p\leq\infty\), which are bandlimited to \(G\) is denoted by \(B_{G,p}\) and by \(B_{T,p}\) for \(G=Q_ T\). Denoting \(\text{sinc}(x)=x^{-1}\sin x\) for \(x\neq 0\) and \(=1\) for \(x=0\) and \(\text{sinc}_ n(x)=\prod_{i=1}^ n \text{sinc}(x_ i)\), \(x\in\mathbb{R}^ n\), then every function \(f\in B_{\sigma,p}\) can be pointwisely represented in the form \(f(x)=\sum_{k\in Z^ n} f(hk)C(x=hk)\), \(\sigma,\delta\in\mathbb{R}_ +^ n\), \(\rho=\sigma+\delta\), \(C(x)=\text{sinc}_ n(\rho x)\text{sinc}_ n(\delta x)\), \(h=\pi/\rho\) (the sampling theorem). The author investigates the \(n\)-width, \(\varepsilon\)-entropy and \(\varepsilon\)- dimension of the set \(SB_{G,p}\) (the functions in \(B_{G,p}\) of \(L_ p\)-norm \(\leq 1\)). It turns out that these quantities have asymptotic orders of the form \((\text{vol }Q_ T)^ s F(\varepsilon,p,q)\), \(s>0\), for \(T\to\infty\). For instance \(\limsup(2T)^{-1} K_ \varepsilon(G,T)\leq(2\pi)^{-n}\text{meas}(G)\), where \(G\) is a bounded, Jordan measurable subset of \(\mathbb{R}^ n\) and \(K_ \varepsilon\) denotes the \(\varepsilon\)-dimension of \(G\), \(0<\varepsilon<1\). The author presents a modification of the sampling theorem whose proof is based on an analogue of Marcinkiewicz' theorem on the equivalence between \(L_ p\)-norms and the discrete \(\ell_ p^ s\)-norms for bandlimited functions and on the truncation \(L_ q^ T\)-approximation. These results are important in information theory and extend results obtained by \textit{V. A. Kotelnikov} [Materials of the First All-Union Conference on Questions on Communications (Moscow, 1933)], \textit{C. E. Shannon} [Proc. IRE 37, 10-21 (1949)], \textit{A. N. Kolmogorov} and \textit{V. N. Tikhomirov} [Usp. Mat. Nauk 14, No. 2(86), 3-86 (1959; Zbl 0133.067)] and others.
scientific article; zbMATH DE number 62705
scientific article