Interpolation and frames in certain Banach spaces of entire functions
The author considers more generally the classical Banach spaces \(E^p(1\leq p\leq \infty)\) consisting of all entire functions of exponential type at most \(\pi\) that belong to \(L^p(-\infty, \infty)\) on the real axis. By virture of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between \(L^2(-\pi, \pi)\) and \(E^2\). When \(p\) is finite, a sequence \(\{\lambda_n\}\) of complex numbers will be called a frame for \(E^p\) provided the inequalities \[ A\| f\| ^p\leq \sum | f(\lambda_n)| ^p \leq B\| f\| ^p \] hold for some positive constants \(A\) and \(B\) and all functions \(f\) in \(E^p\). We say that \(\{\lambda_n\}\) is an interpolating sequence for \(E^p\) if the set of all scalar sequences \(\{f(\lambda_n)\}\), with \(f\in E^p\), coincides with \(l^p\). If in addition \(\{\lambda_n\}\) is a set of uniqueness for \(E^p\), that is, if the relation \(f(\lambda_n)=0(-\infty<n<\infty)\), with \(f\in E^p\), imply that \(f\equiv 0\), then we call \(\{\lambda_n\}\) a complete interpolating sequence. The author shows that every complete interpolating sequence for \(E^p(1<p<\infty)\) remains stable under a very general set of displacements of its elements.
scientific article; zbMATH DE number 1103026