Comptes Rendus
no. 8
Mathematical Analysis
Wiener's lemma for infinite matrices with polynomial off-diagonal decay

Presented by: Yves Meyer

Qiyu Sun1
1 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
Comptes Rendus. Mathématique, Volume 340 (2005) no. 8, pp. 567-570.

In this Note, we give a simple elementary proof to Wiener's lemma for infinite matrices with polynomial off-diagonal decay.

Dans cette Note, nous donnons une preuve elementaire du lemme de Wiener pour les matrices infinies a decroissance polynomiale des termes non-digonaux.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.03.002

Qiyu Sun 1

1 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA 
@article{CRMATH_2005__340_8_567_0,
     author = {Qiyu Sun},
     title = {Wiener's lemma for infinite matrices with polynomial off-diagonal decay},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {567--570},
     publisher = {Elsevier},
     volume = {340},
     number = {8},
     year = {2005},
     doi = {10.1016/j.crma.2005.03.002},
     language = {en},
}
TY  - JOUR
AU  - Qiyu Sun
TI  - Wiener's lemma for infinite matrices with polynomial off-diagonal decay
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 567
EP  - 570
VL  - 340
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2005.03.002
LA  - en
ID  - CRMATH_2005__340_8_567_0
ER  - 
%0 Journal Article
%A Qiyu Sun
%T Wiener's lemma for infinite matrices with polynomial off-diagonal decay
%J Comptes Rendus. Mathématique
%D 2005
%P 567-570
%V 340
%N 8
%I Elsevier
%R 10.1016/j.crma.2005.03.002
%G en
%F CRMATH_2005__340_8_567_0
Qiyu Sun. Wiener's lemma for infinite matrices with polynomial off-diagonal decay. Comptes Rendus. Mathématique, Volume 340 (2005) no. 8, pp. 567-570. doi : 10.1016/j.crma.2005.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.002/

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

[2] R. Balan, P.G. Casazza, C. Heil, Z. Landau, Density, overcompleteness and localization of frames, Preprint, 2004

[3] B.A. Barnes The spectrum of integral operators on Lebesgue spaces, J. Operator Theory, Volume 18 (1987), pp. 115-132

[4] O. Christensen, T. Strohmer, The finite section method and problems in frame theory, J. Approx. Theory, in press

[5] C.K. Chui; W. He; J. Stöckler Nonstationary tight wavelet frames II: unbounded intervals, Appl. Comput. Harmonic Anal., Volume 18 (2005), pp. 25-66

[6] E. Cordero; K. Gröchenig Localization of frames II, Appl. Comput. Harmonic Anal., Volume 17 (2004), pp. 29-47

[7] C. de Boor A bound on the L∞-norm of the L2-approximation by splines in terms of a global mesh ratio, Math. Comput., Volume 30 (1976), pp. 687-694

[8] S. Demko Inverse of band matrices and local convergences of spline projections, SIAM J. Numer. Anal., Volume 14 (1977), pp. 616-619

[9] I.M. Gelfand; D.A. Raikov; G.E. Silov Commutative Normed Rings, Chelsea, New York, 1964

[10] K. Gröchenig Localized frames are finite unions of Riesz sequences, Adv. Comput. Math., Volume 18 (2003), pp. 149-157

[11] K. Gröchenig Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., Volume 10 (2004), pp. 105-132

[12] K. Gröchenig; M. Leinert Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., Volume 17 (2003), pp. 1-18

[13] K. Gröchenig, M. Leinert, Symmetry of matrix algebras and symbolic calculus for infinite matrices, Trans. Amer. Math. Soc., in press

[14] S. Jaffard Properiétés des matrices bien localisées prés de leur diagonale et quelques applications, Ann. Inst. H. Poincaré, Volume 7 (1990), pp. 461-476

[15] R.-Q. Jia; C.A. Micchelli Using the refinement equations for the construction of pre-wavelets II: Powers of two, Curves and Surfaces (Chamonix-Mont-Blanc, 1990), Academic Press, Boston, MA, 1991, pp. 209-246

[16] D.J. Newman A simple proof of Wiener's 1/f theorem, Proc. Amer. Math. Soc., Volume 48 (1975), pp. 264-265

[17] T. Strohmer Rates of convergence for the approximation of shift-invariant systems in ℓ2(Z), J. Fourier Anal. Appl., Volume 5 (2000), pp. 519-616

[18] T. Strohmer Four short stories about Toeplitz matrix calculations, Linear Algebra Appl., Volume 343/344 (2002), pp. 321-344

[19] Q. Sun, in preparation

Cited by Sources:

Comments - Policy