# Construction of sampling and interpolating sequences for multi-band signals. the two-band case

Sergei Avdonin; Anna Bulanova; William Moran

International Journal of Applied Mathematics and Computer Science (2007)

- Volume: 17, Issue: 2, page 143-156
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topAvdonin, Sergei, Bulanova, Anna, and Moran, William. "Construction of sampling and interpolating sequences for multi-band signals. the two-band case." International Journal of Applied Mathematics and Computer Science 17.2 (2007): 143-156. <http://eudml.org/doc/207826>.

@article{Avdonin2007,

abstract = {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.},

author = {Avdonin, Sergei, Bulanova, Anna, Moran, William},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {families of exponentials; observation; multi-band signals; sampling and interpolation; Wiener-Hopf equations; Riesz bases; control; sampling; interpolation},

language = {eng},

number = {2},

pages = {143-156},

title = {Construction of sampling and interpolating sequences for multi-band signals. the two-band case},

url = {http://eudml.org/doc/207826},

volume = {17},

year = {2007},

}

TY - JOUR

AU - Avdonin, Sergei

AU - Bulanova, Anna

AU - Moran, William

TI - Construction of sampling and interpolating sequences for multi-band signals. the two-band case

JO - International Journal of Applied Mathematics and Computer Science

PY - 2007

VL - 17

IS - 2

SP - 143

EP - 156

AB - 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.

LA - eng

KW - families of exponentials; observation; multi-band signals; sampling and interpolation; Wiener-Hopf equations; Riesz bases; control; sampling; interpolation

UR - http://eudml.org/doc/207826

ER -

## References

top- Antonevich A. (1996): Linear Functional Equations. Operator Approach.- Basel, Birkhauser. Zbl0841.47001
- Avdonin S. (1979): On Riesz bases from exponentials in L^2. - Vestnik Leningrad Univ. Math., Vol.7, pp. 203-211. Zbl0427.46016
- 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. Zbl0866.93001
- Avdonin S. and Moran W. (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: Birkhauser, Vol.133, pp.43-51. Zbl0933.35026
- Beaty M. and Dodson M. (1989): Derivative sampling for multiband signals. - Numer. Funct. Anal. Optim., Vol.10, No.9-10, pp.875-898. Zbl0694.94003
- 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. Zbl0777.94003
- Bezuglaya L. and Katsnelson V. (1993): The sampling theorem for functions with limed multi-band spectrum, I. - Z. Anal. Anwendungen, Vol.12, No.3, pp. 511-534. Zbl0786.30019
- Bottcher A., Karlovich Y. and Spitkovsky I. (2002): Convolution Operators and Factorization of Almost Periodic Matrix Functions. - Basel: Birkhauser. Zbl1011.47001
- Dodson M. and Silva A. (1989): An algorithm for optimal regular sampling. - Signal Process., Vol.17, No.2, pp.169-174.
- Higgins J. (1996): Sampling Theory in Fourier and Signal Analysis: Foundations. - Oxford: Clarendon Press. Zbl0872.94010
- 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.
- 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,.
- Kohlenberg A. (1953): Exact interpolation of band-limed functions.- J. Appl. Phys., Vol.24, No.12, pp.1432-1436. Zbl0052.43306
- Lyubarskii Y. and Seip K. (1997): Sampling and interpolating sequences for multiband-limed functions and exponential bases on disconnected sets.- J. Fourier Anal. Appl., Vol.3, No.5, pp.597-615. Zbl0911.42019
- 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. Zbl0847.42002
- 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. Zbl0992.94502
- Peterson K. (1983): Ergodic Theory. - Cambridge: Cambridge University Press.
- Russell D. (1978): Controllability and stabilizability theory for linear partial differential equations. - SIAM Review, Vol.20, No.4, pp.639-739. Zbl0397.93001
- Seip K. (1995): A simple construction of exponential bases in L^2 of the union of several intervals. - Proc. Edinburgh Math. Soc.,Vol.38, No.1, pp.171-177. Zbl0826.30003
- Spitkovsky I. (2006): Personal communication

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.