Geodesic
Construction of sampling and interpolating sequences for multi-band signals. The two-band case
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 2, pp. 143-156.

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Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.
Keywords: sampling and interpolation, multi-band signals, Riesz bases, families of exponentials, Wiener–Hopf equations, control, observation
Mots-clés : próbkowanie, interpolacja, bazy Riesza, równanie Wienera-Hopfa, sterowanie, obserwacja 
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Avdonin, S.; Bulanova, A.; Moran, W. Construction of sampling and interpolating sequences for multi-band signals. The two-band case. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 2, pp. 143-156. http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_2_a0/

[1] Antonevich A. (1996): Linear Functional Equations. Operator Approach. -Basel, Birkhäuser.

[2] Avdonin S. (1979): On Riesz bases from exponentials in L2. - Vestnik Leningrad Univ. Math., Vol. 7, pp. 203-211.

[3] Avdonin S. and Ivanov S. (1995): Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. - New York: Cambridge University Press.

[4] Avdonin S. and MoranW. (1999): Sampling and interpolation of functions with multi-band spectra and controllability problems, In: Optimal Control of Partial Differential Equations, (K.H. Hoffmann, G. Leugering and T. F., Eds.), Basel: Birkhäuser, Vol. 133, pp. 43-51.

[5] Beaty M. and Dodson M. (1989): Derivative sampling for multiband signals. - Numer. Funct. Anal. Optim., Vol. 10, No. 9-10, pp. 875-898.

[6] Beaty M. and Dodson M. (1993): The distribution of sampling rates for signals with equally wide, equally spaced spectral bands. - SIAM J. Appl. Math., Vol. 53, No. 3, pp. 893-906.

[7] Bezuglaya L. and Katsnelson V. (1993): The sampling theorem for functions with limited multi-band spectrum, I. - Z. Anal. Anwendungen, Vol. 12, No. 3, pp. 511-534.

[8] Böttcher A., Karlovich Y. and Spitkovsky I. (2002): Convolution Operators and Factorization of Almost Periodic Matrix Functions. -Basel: Birkhäuser.

[9] Dodson M. and Silva A. (1989): An algorithm for optimal regular sampling. - Signal Process., Vol. 17, No. 2, pp. 169-174.

[10] Higgins J. (1996): Sampling Theory in Fourier and Signal Analysis: Foundations. -Oxford: Clarendon Press.

[11] Hruščev S., Nikol'skii N. and Pavlov B. (1981): Unconditional bases of exponentials and reproducing kernels, In: Complex Analysis and Spectral Theory (V.P Havin and N.K. Nikol'ski, Eds.), Lecture Notes Math., Vol. 864, pp. 214-335.

[12] Katsnelson V. (1996): Sampling and interpolation for functions with multi-band spectrum: The mean-periodic continuation method, In: Wiener-Symposium (Grossbothen, 1994) Synerg. Syntropie Nichtlineare Syst., Vol. 4, Leipzig: Verlag Wiss. Leipzig, pp. 91-132,.

[13] Kohlenberg A. (1953): Exact interpolation of band-limited functions. - J. Appl. Phys., Vol. 24, No. 12, pp. 1432-1436.

[14] Lyubarskii Y. and Seip K. (1997): Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets. - J. Fourier Anal. Appl., Vol. 3, No. 5, pp. 597-615.

[15] Lyubarskii Y. and Spitkovsky I. (1996): Sampling and interpolation for a lacunary spectrum. -Proc. Royal. Soc. Edinburgh, Vol. 126 A, No. 1, pp. 77-87.

[16] Moran W. and Avdonin S. (1999): Sampling of multi-band signals. - Proc. 4-th Int. Congress Industrial and Applied Mathematics, Edinburgh, Scotland, pp. 163-174.

[17] Peterson K. (1983): Ergodic Theory.-Cambridge: Cambridge University Press.

[18] Russell D. (1978): Controllability and stabilizability theory for linear partial differential equations. - SIAMReview, Vol. 20, No. 4, pp. 639-739.

[19] Seip K. (1995): A simple construction of exponential bases in L of the union of several intervals.-Proc. Edinburgh Math. Soc., Vol. 38, No. 1, pp. 171-177.

[20] Spitkovsky I. (2006): Personal communication.