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2001 | 11 | 4 | 803-820

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Ingham-type inequalities and Riesz bases of divided differences

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We study linear combinations of exponentials e^{iλ_nt} , λ_n ∈ Λ in the case where the distance between some points λ_n tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L^2 (0,T). Here we prove that if the upper uniform density of Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{iλ_nt}} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from {e^{iλ_nt}} a suitable collection of functions e^{iλ_nt}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.








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  • Department of Mathematical Sciences, University of Alaska Fairbanks, P.O. Box 756660, Fairbanks, AK 99775-6660, U.S.A.
  • Department of Mathematics and Statistics, Flinders University of South Australia, GPO Box 2100, Adelaide 5001, Australia


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