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2001 | 11 | 4 | 803-820
Tytuł artykułu

Ingham-type inequalities and Riesz bases of divided differences

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
11
Numer
4
Strony
803-820
Opis fizyczny
Daty
wydano
2001
otrzymano
2001-06-01
poprawiono
2001-09-01
Twórcy
  • 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
Bibliografia
  • Avdonin S.A. and Ivanov S.A. (2000): Levin-Golovin theorem for the Sobolev spaces. — Matemat. Zametki, Vol.68, No.2, pp.163–172 (in Russian); English transl. in Math. Notes, Vol.68, No.2, pp.145–153.
  • Avdonin S.A. and Ivanov S.A. (2001): Exponential Riesz bases of subspaces and divided differences. — Algebra i Analiz., Vol.13, No.3, pp.1–17 (in Russian); English transl. in St. Petersburg Math. Journal, Vol.13, No.3
  • Avdonin S.A. and Joó I. (1988): Riesz bases of exponentials and sine type functions. — Acta Math. Hung., Vol.51, No.1–2, pp.3–14.
  • Avdonin S. and Moran W. (2001): Simultaneous control problems for systems of elastic strings and beams. — Syst. Contr. Lett., Vol.44, No.2, pp.147–155.
  • Avdonin S. and Tucsnak M. (2001): Simultaneous controllability in short time of two elastic strings. — ESAIM Contr. Optim. Calc. Var., Vol.6, pp.259–273.
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  • Baiocchi C., Komornik V. and Loreti P. (2000): Généralisation d’un théorème de Beurling et application a la théorie du contrôle. — C. R. Acad. Sci. Paris, Vol.330, Série I, pp.281– 286.
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  • Jaffard S., Tucsnak M. and Zuazua E. (1997): On a theorem of Ingham. — J. Fourier Anal. Appl., Vol.3, No.5, pp.577–582.
  • Jaffard S., Tucsnak M. and Zuazua E. (1998): Singular internal stabilization of the wave equation. — J. Diff. Eq., Vol.145, No.1, pp.184–215.
  • Khrushchev S.V., Nikol’skiĭ N.K. and Pavlov B.S. (1981): Unconditional bases of exponentials and reproducing kernels. —Lecture Notes in Math., Vol.864, pp.214–335, Berlin/Heidelberg: Springer.
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Bibliografia
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