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.
@article{bwmeta1.element.bwnjournal-article-amcv11i4p803bwm,
author = {Avdonin, Sergei and Moran, William},
title = {Ingham-type inequalities and Riesz bases of divided differences},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {11},
year = {2001},
pages = {803-820},
zbl = {1031.93098},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i4p803bwm}
}
Avdonin, Sergei; Moran, William. Ingham-type inequalities and Riesz bases of divided differences. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 803-820. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i4p803bwm/
[000] 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. | Zbl 0989.46017
[001] 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
[002] 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. | Zbl 0645.42027
[003] 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. | Zbl 0986.93037
[004] 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. | Zbl 0995.93036
[005] 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. | Zbl 0924.42022
[006] 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. | Zbl 0964.42019
[007] 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.
[008] Butkovsky A.G. (1965): Theory of Optimal Control of Distributed Parameter Systems. — Moscow: Nauka (Russian); English transl. 1969.
[009] Cassels J.W.S. (1965): An Introduction to Diophantine Approximation. — Cambridge: Cambridge University Press. | Zbl 0077.04801
[010] 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. | Zbl 0909.35085
[011] Garnett J.B. (1981): Bounded Analytic Functions. — New York: Academic Press. | Zbl 0469.30024
[012] 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. | Zbl 0853.93018
[013] Isaacson E. and Keller H.B. (1966): Analysis of Numerical Methods. — New York: Wiley. | Zbl 0168.13101
[014] Jaffard S., Tucsnak M. and Zuazua E. (1997): On a theorem of Ingham. — J. Fourier Anal. Appl., Vol.3, No.5, pp.577–582. | Zbl 0939.42004
[015] 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. | Zbl 0920.35029
[016] 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.
[017] Lagnese J.E. and Lions J.-L. (1988): Modelling, Analysis and Control of Thin Plates. — Paris: Masson. | Zbl 0662.73039
[018] Lions J.-L. (1988): Controlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Volume 1. — Paris: Masson. | Zbl 0653.93002
[019] Lyubarskii Yu. and Seip K. (2000): Weighted Paley-Wiener spaces. — (submitted).
[020] Lopes A. and Zuazua E. (1998): Null controllability of the 1-d heat equation as limit of the controllability of dissipative wave equations. — C. R. Acad. Sci. Paris Ser. I Math., Vol.327, No.8, pp.753–758. | Zbl 0918.93005
[021] 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.
[022] Micu S. and Zuazua E. (1997): Boundary controllability of a linear hybrid system arising in the control of noise. — SIAM J. Contr. Optim., Vol.35, No.5, pp.1614–1637. | Zbl 0888.35017
[023] Nikol’skiĭ N.K. (1986): A Treatise on the Shift Operator. — Berlin: Springer.
[024] Pavlov B.S. (1979): Basicity of an exponential system and Muckenhoupt’s condition. — Soviet Math. Dokl., Vol.20, pp.655–659. | Zbl 0429.30004
[025] Paley R.E.A.C. and Wiener N. (1934): Fourier Transforms in the Complex Domain. — New York: AMS Coll. Publ., Vol.XIX. | Zbl 0011.01601
[026] Russell D.L. (1978): Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions. — SIAM Review, Vol.20, No.4, pp.639– 739. | Zbl 0397.93001
[027] Russell D.L. (1986): The Dirichlet-Neumann boundary control problem associated with Maxwell’s equations in a cylindrical region. — SIAM J. Contr. Optim., Vol.24, No.2, pp.199–229. | Zbl 0594.49026
[028] Seip K. (1995): On the connection between exponential bases and certain related sequences in L2 (−π, π). — J. Funct. Anal., Vol.130, No.1, pp.131–160. | Zbl 0872.46006
[029] Shilov G.E. (1965): Mathematical Analysis. The Second Additional Course. — Moscow: Nauka (in Russian).
[030] Tucsnak M. and Weiss G. (2000): Simultaneous exact controllability and some applications. — SIAM J. Contr. Optim., Vol.38, No.5, pp.1408–1427. | Zbl 0982.93021
[031] Ullrich D. (1980): Divided differences and system of nonharmonic Fourier series. — Proc. Amer. Math. Soc., Vol.80, No.1, pp.47–57. | Zbl 0448.42009
[032] Young R.M. (1980): An Introduction to Nonharmonic Fourier Series. — New York: Academic Press. | Zbl 0493.42001