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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Journal of Fourier A...arrow_drop_down
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Journal of Fourier Analysis and Applications
Article . 1997 . Peer-reviewed
License: Springer TDM
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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
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Article . 1997
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On a theorem of Ingham

Authors: Jaffard, S.; Tucsnak, M.; Zuazua, E.;

On a theorem of Ingham

Abstract

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.

Keywords

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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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
19
Average
Top 10%
Average
Green