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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.
  • Baiocchi C., Komornik V. and Loreti P. (1999): Ingham type theorem and applications to control theory. — Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat., Vol.8, No.2(1), pp.33–63.
  • 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.
  • Beurling A. (1989): Balayage of Fourier-Stieltjes transforms, In: The Collected Works of Arne Beurling, Vol.2 Harmonic Analysis (L. Carleson, P. Malliavin, J. Neuberger, J. Wermer, Eds.). — Boston: Birkhäuser.
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  • Cassels J.W.S. (1965): An Introduction to Diophantine Approximation. — Cambridge: Cambridge University Press.
  • Castro C. and Zuazua E. (1998): Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass. — SIAM J. Contr. Optim., Vol.36, No.5, pp.1576–1595.
  • Garnett J.B. (1981): Bounded Analytic Functions. — New York: Academic Press.
  • Hansen S. and Zuazua E. (1995): Exact controllability and stabilization of a vibrating string with an interior point mass. — SIAM J. Contr. Optim., Vol.33, No.5, pp.1357–1391.
  • Isaacson E. and Keller H.B. (1966): Analysis of Numerical Methods. — New York: Wiley.
  • 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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  • Minkin A.M. (1991): Reflection of exponents, and unconditional bases of exponentials. — Algebra i Analiz, Vol.3, No.5, pp.109–134 (in Russian); English transl. in St. Petersburg Math. J., Vol.3 (1992), No.5, pp.1043–1068.
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  • Nikol’skiĭ N.K. (1986): A Treatise on the Shift Operator. — Berlin: Springer.
  • Pavlov B.S. (1979): Basicity of an exponential system and Muckenhoupt’s condition. — Soviet Math. Dokl., Vol.20, pp.655–659.
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Bibliografia

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