
doi: 10.1007/bf02648885
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.
nonharmonic Fourier series, control of PDEs, Classical almost periodic functions, mean periodic functions, Completeness of sets of functions in one variable harmonic analysis, Nontrigonometric harmonic analysis involving wavelets and other special systems, Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization, Existence theories for optimal control problems involving partial differential equations, Article, Riesz basis, 510.mathematics, Spline approximation, spline functions
nonharmonic Fourier series, control of PDEs, Classical almost periodic functions, mean periodic functions, Completeness of sets of functions in one variable harmonic analysis, Nontrigonometric harmonic analysis involving wavelets and other special systems, Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization, Existence theories for optimal control problems involving partial differential equations, Article, Riesz basis, 510.mathematics, Spline approximation, spline functions
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