On irregular sampling and interpolation in Bernstein spaces (Q2423220)

The main results in this paper deal with the sharpness of interpolation- and sampling-constants and are given in the Theorems 1 and 2. The first one was partly announced by the authors in [C. R., Math., Acad. Sci. Paris 353, No. 1, 47--50 (2015; Zbl 1309.42030)]: Theorem 1. Let \(\Lambda\subset\mathbb{R}\) be a u.d. set. (i) If \(D{-}(\lambda)=1\), then \[ K_s(\Lambda, B_{\sigma})\geq C\log{\frac{\pi}{\pi-\sigma}},\ 0<\sigma <\pi. \] (ii) If \(D{+}(\lambda)=1\), then \[ K_i(\Lambda, B_{\sigma})\geq C\log{\frac{\sigma}{\sigma-\pi}},\ \pi<\sigma <2\pi. \] In the second result, \(\Lambda\subset\mathbb{T}\) is a finite set. Theorem 2. Assume \(\Lambda\subset\mathbb{T}\). (i) If \(\#\Lambda\geq n+1\), then \[ K_s(\Lambda, P_n)\geq C\log{\frac{n}{\#\Lambda-n}}. \] (ii) If \(\#\Lambda\leq n+1\), then \[ K_i(\Lambda, P_n)\geq C\log{\frac{n}{n+2-\#\Lambda}}. \] The concepts used are: -- the \textit{Bernstein space} \(B_{\sigma}\): the space of entire functions of exponential type \(\sigma>0\), bounded on \(\mathbb{R}\), equipped with the uniform norm, -- \(\Lambda\subset\mathbb{R}\) is a \textit{sampling set\/} for \(B_{\sigma}\) if there is a constant \(K\) with \[ \|f\|\leq K\|f|_{\Lambda}\|\hbox{ for every }f\in B_{\sigma}. \] -- the \textit{sampling norm} is \(\|f|_{\Lambda}\|=\sup_{\lambda\in\Lambda}\,|f(\lambda)|\). -- the \textit{sampling constant} is \[ K_s(\Lambda,B_{\sigma})=\sup_{f\in B_{\sigma},f\not= 0}\,\frac{\|f\|}{\|f|_{\Lambda}\|}. \] -- a discrete set is an \textit{interpolation set} for \(B_{\sigma}\) if for every sequence \(\{c(\lambda)\}\in \ell^{\infty}(\Lambda)\) there exists an \(f\in B_{\sigma}\) with \(f(\lambda)=c(\lambda),\ \lambda\in\Lambda\). -- the set \(\Lambda\) is said to be \textit{uniformly discrete} (u.d.) if \[ \inf_{\lambda,\lambda'\in\Lambda,\lambda\not=\lambda'}\,|\lambda-\lambda'|>0. \] -- the \textit{interpolation constant} \(K_i(\Lambda,B_{\sigma})\) is the minimal constant \(K\), such that \(\|f\| \leq K\|f|_{B_{\sigma}}\|\) for all \(f\in B_{\sigma}\) that interpolate in a bounded sequence. -- for Theorem 2 more definitions are in order. After this, the proofs take 10 pages of the paper. The list of references contains 8 items.