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Sampling and interpolation for a lacunary spectrum

Published online by Cambridge University Press:  14 November 2011

Yu. Lyubarskii  and
I. Spitkovsky
Yu. Lyubarskii
Affiliation:
Institute for Low Temperature Physics and Engineering, Academy of Science of Ukraine, 47, Lenin pr., Kharkov, 310164, Ukraine e-mail: [email protected]
I. Spitkovsky
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, U.S.A. e-mail; [email protected]
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Abstract

We reduce the problem of constructing a sampling and interpolating set for the space of functions with limited multi-band spectra to a problem of invertibility of certain convolution operators on a system of intervals, and obtain an example of such a set located in a horizontal strip along the real axis. We also study the question of sampling of a signal with two-banded spectra via its values at the union of two arithmetic progressions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 1 , 1996 , pp. 77 - 87
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Bezuglaya, L. and Katsnelson, V.. The sampling theorem for functions with limited multi-band spectrum. I. Z. Anal. Anwendungen 12 (1993), 511–34.CrossRefGoogle Scholar
2Higgins, J. R.. Five short stories about the cardinal series. Bull. Amer. Math. Soc. 12 (1985), 45–89.Google Scholar
3Higgins, J. R.. Sampling theory for Paley-Wiener spaces in the Riesz basis setting. Proc. Roy. Irish Acad. (to appear).Google Scholar
4Karlovich, Yu. I. and Spitkovsky, I. M.. Factorization of almost periodic matrix functions and the Noether theory for certain classes of equations of convolution type. Izv. Akad. Nauk SSSR, Ser. Mat. 53 (1989), 275–308 (in Russian); English translation in Mathematics of the USSR, Izvestiya 34 (1990), 281-3 6.Google Scholar
5Kholenben, A.;. Exact interpolation of band-limited functions. J. Appl. Phys. 24 (1953), 1432–6.Google Scholar
6Khrushchev, S. V., Nikolskii, N. K. and Pavlov, B. S.. Unconditional bases of exponentials and reproducing kernels. In Complex Analysis and Spectral Theory, eds Havin, V. P. and Nikolskii, N. K., Lecture Notes in Mathematics 864, pp. 214–335 (Berlin: Springer, 1981).Google Scholar
7Landau, H. J.. Necessary density conditions for sampling and interpolation of certain entire functions. Ada Math. 117 (1967), 37–52.Google Scholar
8Levin, B. Ya.. On bases of exponential functions in L2(–π, π). Khar'kov. Gos. Univ. Uchen. Zap 115 (=Zap. Mat. Otdel. Fiz.-Mat. Fak. i Kharkov Mat. Obsch. (4) 27 (1961), 39–48) (in Russian).Google Scholar
9Levin, B. Ya.. Interpolation by entire functions of exponential type. Mat. Fiz. i Funktsional. Anal. Vyp. 1 (1969), 136–46 (in Russian).Google Scholar
10Paley, R. and Wiener, N.. Fourier Transform in the Complex Domain (New York: American Mathematical Society, 1934).Google Scholar
11Pavlov, B. S.. Basicity of an exponential system and Muckenhoupt's condition. Soviet Math. Dokl. 20 (1979), 655–9.Google Scholar
12Seip, K.. A simple construction of exponential bases in L2 of the union of several intervals. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar