On a theorem of Ingham
The problem of determining under which conditions a sequence of the form \((e^{i \lambda_n t})\) is a Riesz basis has been settled by \textit{S. V. Hrushev, N. K. Nikol'skii} and \textit{B. S. Pavlov} [Lect. Notes Math. 864, 214-335 (1981; Zbl 0466.46018)], but their condition is difficult to verify. The main result of this paper is the following: Theorem. Let \(\lambda_n\) be a strictly increasing sequence satisfying \[ \lambda_{n+2} - \lambda_n \geq \delta > 0. \] Let \(A = \{n : \lambda_{n+1} - \lambda_n \leq \delta/2\}\) and \(B = \{n : n \notin A\) and \(n-1 \notin A\}\). For any \(T > 6\sqrt{6}/\delta\), there exists \(C>0\) such that \[ \int_{-T}^{T} \left |\sum a_n e^{i \lambda_n t} \right |^2 dt \geq C \sum_{n \in A} \bigg ( |a_n|^2 + |a_{n+1}|^2 \bigg) |\lambda_{n+1} - \lambda_n|^2 + |a_n+a_{n+1}|^2 + C \sum_{n \in B} |a_n|^2. \] This result can be used to determine whether a sequence of exponentials is a Riesz basis, and to find lower Riesz bounds. However, the initial motivation for the result arose in the context of control of partial differential equations [\textit{S. Jaffard, M. Tucsnak} and \textit{E. Zuazua}, J. Differ. Equations 145, No. 1, 184-215 (1998; Zbl 0920.35029)]. The proof of the theorem uses properties of \(B\)-splines.
scientific article; zbMATH DE number 1103022