Sampling in a weighted Sobolev space

Acala, Nestor G.; Reyes, Noli N.

doi:10.1016/j.crma.2012.10.028

Mathematical Analysis/Theory of Signals

Sampling in a weighted Sobolev space
[Échantillonage dans un espace de Sobolev avec poids]

Présenté par : Yves Meyer

Acala, Nestor G.¹ ; Reyes, Noli N.¹

¹ University of the Philippines – Diliman, Institute of Mathematics, Quezon City, 1101, Philippines

Comptes Rendus. Mathématique, Tome 350 (2012) no. 21-22, pp. 941-944.

Résumé
Abstract

Nous démontrons que toute fonction f dans un certain espace de Sobolev avec poids est complètement determinée par un échantillon ${f (t_{n})}_{n \in Z} \cup {\hat{f} (λ_{k})}_{k \in Z}$ sur des convenables suites croissantes ${t_{n}}_{n \in Z}$ et ${λ_{n}}_{n \in Z}$ , tendant vers ±∞ lentement, quand $n \to \pm \infty$ .

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${f (t_{n})}_{n \in Z} \cup {\hat{f} (λ_{k})}_{k \in Z}$ along appropriate slowly increasing sequences ${t_{n}}_{n \in Z}$ and ${λ_{n}}_{n \in Z}$ tending to ±∞ as $n \to \pm \infty$ .

Reçu le : 2011-11-10
Accepté le : 2012-10-29
Publié le : 2012-11-01

DOI : 10.1016/j.crma.2012.10.028

Affiliations des auteurs :

Acala, Nestor G. ¹ ; Reyes, Noli N. ¹

¹ University of the Philippines – Diliman, Institute of Mathematics, Quezon City, 1101, Philippines

@article{CRMATH_2012__350_21-22_941_0,
     author = {Acala, Nestor G. and Reyes, Noli N.},
     title = {Sampling in a weighted {Sobolev} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {941--944},
     publisher = {Elsevier},
     volume = {350},
     number = {21-22},
     year = {2012},
     doi = {10.1016/j.crma.2012.10.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.10.028/}
}

TY  - JOUR
AU  - Acala, Nestor G.
AU  - Reyes, Noli N.
TI  - Sampling in a weighted Sobolev space
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 941
EP  - 944
VL  - 350
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.10.028/
DO  - 10.1016/j.crma.2012.10.028
LA  - en
ID  - CRMATH_2012__350_21-22_941_0
ER  -

%0 Journal Article
%A Acala, Nestor G.
%A Reyes, Noli N.
%T Sampling in a weighted Sobolev space
%J Comptes Rendus. Mathématique
%D 2012
%P 941-944
%V 350
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.10.028/
%R 10.1016/j.crma.2012.10.028
%G en
%F CRMATH_2012__350_21-22_941_0

Acala, Nestor G.; Reyes, Noli N. Sampling in a weighted Sobolev space. Comptes Rendus. Mathématique, Tome 350 (2012) no. 21-22, pp. 941-944. doi : 10.1016/j.crma.2012.10.028. http://www.numdam.org/articles/10.1016/j.crma.2012.10.028/

Bibliographie
Cité par

[1] Aldroubi, A.; Gröchenig, K. Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., Volume 43 (2001), pp. 585-620

[2] Butzer, P.L.; Ferreira, P.J.S.G.; Higgins, J.R.; Saitoh, S.; Schmeisser, G.; Stens, R.L. Interpolation and sampling: E.T. Whittaker, K. Ogura and their followers, J. Fourier Anal. Appl., Volume 17 (2010), pp. 320-354

[3] Donoho, D.L.; Stark, P.B. Uncertainty principles and signal recovery, SIAM J. Appl. Math., Volume 49 (1990), pp. 906-931

[4] Feichtinger, H.G.; Gröchenig, K. Theory and practice of irregular sampling, Wavelets: Mathematics and Applications, Stud. Adv. Math., CRC, Boca Raton, 1994, pp. 305-363

[5] Gröchenig, K. Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type, Math. Comp., Volume 68 (1999), pp. 749-765

[6] Havin, V.; Jöricke, B. The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994

[7] Meyer, Y. Ondelettes et Opérateurs, Hermann, Paris, 1990

[8] Talvila, E. Rapidly growing Fourier integrals, Amer. Math. Month., Volume 108 (2001), pp. 636-641

[9] Walker, W.J. Zeros of the Fourier transform of a distribution, J. Math. Anal. Appl., Volume 154 (1989), pp. 77-91

Cité par Sources :