Irregular sampling of bandlimited \(L^ p\)-functions
For \(1\leq p\leq\infty\) and \(\beta>0\) denote by \(B^ p_ \beta\) the space of all entire functions \(f\) which are in \(L^ p(R)\) when restricted to \(R\), and satisfy the inequality \(| f(x+iy)|\leq\sup_{u\in R}| f(u)| e^{\beta| y|}\), \(x+iy\in C\). The functions in \(B^ p_ \beta\) are called bandlimited to \([-\beta,\beta]\), since their distributional Fourier transforms vanish outside \([-\beta,\beta]\). A sequence \(\{t\}=\{t_ n\}_{n\in Z}\) of real numbers is called equidistant perturbed with respect to \(Z\) if there exists \(L\geq 0\) such that \(| t_ n-n|\leq L\), \(n\in Z\). For \(s\in R\) and \(z\in C\), let \(g(s,z)=1-z/s\), for \(s\neq 0\), and \(g(0,z)=z\). Let also \(G(z)=G\bigl(\{t\},z\bigr)=g(t_ 0,z)\prod^ \infty_{k=1}g(t_ k,z)\cdot g(t_{-k},z)\), \(z\in C\). The paper is concerned with sampling a function \(f\in B^ p_ \beta\), knowing its values at the points \(t_ n\). For instance \(f(z)=\sum^ \infty_{n=- \infty}f(n)\cdot\psi_ n(z)\), \(\psi_ n(z)=G(z)/\bigl(G'(t_ n)\cdot,(t-t_ n)\bigr)\) for \(z\neq t_ n\) and \(\psi(t_ n)=1\), provided \(L<1/4\), if \(1\leq p\leq 2\) and \(L<1/2p\) if \(2\leq p<\infty\). If \(L\leq 1/4\), for \(p=1\), and \(L<1/4p\), for \(1<p<\infty\), then the convergence is absolute and uniform on every bounded subset of \(C\). For \(t_ n=n\) and \(G(z)=\pi^{-1}\cdot\sin\pi z\) one obtains the classical Whittaker-Shanon-Kotel'nikov sampling theorem: \[ f(z)=\sum^ \infty_{n=-\infty}f(n)\cdot{\sin\pi(z-n)\over\pi(z-n)},\;z\in C. \] Section 6 is concerned with derivative sampling and oversampling. A unified approach to all types of nonuniform samplings considered in this paper, can be found in the author's doctoral thesis [Aachen: RWTH Aachen, Math.-Naturwiss. Fak./FB 1, Diss. 111 S. (1991; Zbl 0725.65140)]. The most general result obtained in this thesis is presented at the end of the paper.
scientific article; zbMATH DE number 205430