Abstract
We employ spectral properties of the infinitesimal generator of the left translation semigroup to obtain, a short, constructive proof of existence of the best 𝕃2(0, ∞)-approximation with exponential sums. The problem has important implications in control theory. Some numerical aspects and possible generalizations of the proposed construction of the best approximant are discussed. A simple, but nontrivial example illustrating the results is provided.
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References
D.W. Kammler, R.J. McGlinn, A bibliography for approximation with exponential sums, Journal of Computational & Applied Mathematics 4 (1978), 167–173.
D.W. Kammler, Approximation with sums of exponentials in L p (0,∞), Journal of Approximation Theory 16 (1976), 384–408.
P. Fuhrmann, Linear Systems and Operators in Hilbert Space, McGraw-Hill, N.Y., 1981.
K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.
N.K. Nikolskii, Treatise on the Shift Operator, Springer, 1985.
S. Kus, On the connection between approximation in the space L 2 (-t∞,+∞) and interpolation, Demonstratio Mathematica 12 (1979), 463–468.
P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, Vol.1: One-dimesional Theory, Birkhäuser, Basel, 1971.
H.B. Dwight, Tables of Integrals and Other Mathematical Data, McMillan, N.Y., 1961.
V. Erokhin, On the best approximation of analytic functions with free poles, Doklady AN SSSR 128 (1959), 29–32, (in Russian).
K. Glover, J. Lam, J. Partington, Rational approximation of a class of infinite — dimensional systems: the L 2 -case, to appear in Journal of Approximation Theory, Reprint CUED/F-INFENG/TR.20., Cambridge University, 1989.
J.M. Anderson, A note on a basis problem, Proceedings of AMS 51 (1975), 330–334.
M.M. Dzharbashian, Basisness of some biorthogonal systems and the solution of interpolation problem in H + P, Doklady AN SSSR 234 (1977), 517–520, (in Russian).
G. Ruckebush, Sur l’approximation rationelle des filtres, Raport N°35, CMA Ecole Polytechnique, 1978.
L. Baratchart, Existence and generic properties of L 2-approximants for linear systems, IMA Journal of Mathematical Control & Information 3 (1986), 89–101.
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© 1991 Springer Basel AG
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Grabowski, P. (1991). On a problem of the best L2-approximation with exponential sums. In: Desch, W., Kappel, F., Kunisch, K. (eds) Estimation and Control of Distributed Parameter Systems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 100. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6418-3_8
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DOI: https://doi.org/10.1007/978-3-0348-6418-3_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-2676-0
Online ISBN: 978-3-0348-6418-3
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