Abstract
We reduce the problem of constructing an exponential Riesz basis in L 2 on the union of several intervals and the equivalent problem of constructing a sampling and interpolating set for the space of functions with limited multi-band spectra to a controllability problem for a model dynamical system. As a model we consider the wave equation with piecewise constant density and boundary control supported on the same union of intervals. For the case of two intervals we construct a controllable system and, as a consequence, a Riesz basis of exponentials produced by the spectrum of the system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S.A. Avdonin, M.I. Belishev and S.A. Ivanov, Boundary control and matrix inverse problem for the equation u tt − u xx + V(x)u = 0, Math. USSR Sbornik, 72 (1992), 287–310.
S.A. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient condition for the observation, control and observation of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024–1065.
L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum, I, Z. Anal. Anwendungen, 12 (1993), 511–534.
J.R. Higgins, Sampling theory for Pale-Wiener spaces in the Riesz basis setting, Proc. Royal Irish Acad., Sect. A, 94 (1994), 219–236.
S.V. Hruscev, N.K. Nikol’skii and B.S. Pavlov, Unconditional bases of exponentials and reproducing kernels, Complex Analysis and Spectral Theory, Lecture Notes Math., 864, (1981), 214–335, Springer-Verlag, Berlin Heidelberg.
V.E. Katsnelson, Sampling and interpolation for functions with multi-band spectrum: the mean-periodic continuation method, Wiener-Symposium (Grossbothen, 1994), 91-132, Synerg. Syntropie Nichtlineare Syst., 4, Verlag Wiss. Leipzig, Leipzig, 1996.
I.S. Katz and M.G. Krein, On spectral functions of a string, Appendix in: F.A. Atkinson, Discrete and Continuous Boundary Problems (Russian edition), Mir, Moscow, 1968.
A. Kohlenberg, Exact interpolation of band-limited functions, J. Appl. Phys., 24 (1953), 1432–1436.
H.J. Landau, Necessary density conditions for sampling and interpolation of certain intire functions, Acta Math., 117, 37–52.
Yu. Lyubarskii and K. Seip, Sampling and interpolating sequences for multibandlimited functions and exponential bases on disconnected sets, J. Fourier Analysis Appl., 3 (1997), 597–615.
Yu. Lyubarskii and I. Spitkovsky, Sampling and interpolation for a lacunary spectrum, Proc. Royal. Soc. Edinburgh, 126 A (1996), 77–87.
M.A. Naimark, Linear Differential Operators, v. 1, Ungar, New York, 1967.
B.S. Pavlov, Basicity of an exponential system and Muckenhoupt’s condition, Soviet Math. Dokl., 20 (1979), 655–659.
D.L. Russell, Controllability and stabilizability theory for linear partial differential equations, SIAM Review, 20 (1978), 639–739.
K. Seip, A simple construction of exponential bases in L 2 of the union of several intervals, Proc. Edinburgh Math. Soc., 38 (1995), 171–177.
A.N. Tikhonov and A.A. Samarskii, Equations of Mathematical Physics, McMillan, New York, 1963.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Avdonin, S., Moran, W. (1999). Sampling and Interpolation of Functions with Multi-Band Spectra and Controllability Problems. In: Hoffmann, KH., Leugering, G., Tröltzsch, F., Caesar, S. (eds) Optimal Control of Partial Differential Equations. ISNM International Series of Numerical Mathematics, vol 133. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8691-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8691-8_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9731-0
Online ISBN: 978-3-0348-8691-8
eBook Packages: Springer Book Archive