Invertibility of the Gabor frame operator on the Wiener amalgam space

Journal of Approximation Theory

Volumn 153, Issue 2, 2008, Pages 212-224

  Krishtal, Ilya A ^a   Okoudjou, Kasso A ^b  

^a NORTHERN ILLINOIS UNIVERSITY   (United States)

^b Bldg 142   (United States)

Author keywords

Gabor frame; Walnut's representation; Wiener amalgam space; Wiener's lemma

Indexed keywords

EID: 49449108600     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2008.03.004     Document Type: Article

References (37)

1. R. Balan, A noncommutative Wiener lemma and a faithful tracial state on Banach algebra of time-frequency operators, Trans. Amer. Math. Soc. 360(7) (2008) 3921-3941.
2. R. Balan, I. Krishtal, An almost periodic noncommutative Wiener's Lemma, preprint.
3. Baskakov A.G. Wiener's theorem and asymptotic estimates for elements of inverse matrices. Funct. Anal. Appl. 24 (1990) 222-224
4. Baskakov A.G. Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis. (Russian) Sibirsk. Mat. Zh. 38 1 (1997) 14-28 (translation in Siberian Math. J. 38(1) (1997) 10-22)
5. Baskakov A.G. Estimates for the elements of inverse matrices, and the spectral analysis of linear operators. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 6 (1997) 3-26 (translation in Izv Math. 61(6) (1997) 1113-1135)
6. Bölcskei H. A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported. J. Fourier Anal. Appl. 5 5 (1999) 409-419
7. Bölcskei H., and Janssen A.J.E.M. Gabor frames, unimodularity, and window decay. J. Fourier Anal. Appl. 6 3 (2000) 255-276
8. de Boor C., DeVore R., and Ron A. The structure of shift invariant spaces and applications to approximation theory. J. Funct. Anal. 119 (1994) 37-78
9. Casazza P.G., Christensen O., and Janssen A.J.E.M. Weil-Heisenberg frames, translation invariant systems and the Walnut representation. J. Funct. Anal. 180 1 (2001) 85-147
10. Casazza P.G., and Lammers M.C. Bracket products for Weyl-Heisenberg frames. In: Feichtinger H.G., and Strohmer T. (Eds). Advances in Gabor Analysis. Applied Numeric Harmon. Anal (2003), Birkhäuser, Boston, MA 71-98
11. Christensen O. Pairs of dual Gabor frame generators with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20 3 (2006) 403-410
12. Daubechies I. Ten Lectures on Wavelets (1992), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
13. Dunford N., and Schwartz J.T. Linear operators. Part I. General Theory. Wiley Classics Library. A Wiley-Interscience Publication (1988), Wiley, New York
14. B. Farrell, T. Strohmer, Wiener's lemma for the Heisenberg group and a class of pseudodifferential operators, J. Operator Theory, to appear.
15. Feichtinger H.G. On a new Segal algebra. Monatsh. Math. 92 4 (1981) 269-289
16. H.G. Feichtinger, Banach Convolution Algebras of Wiener Type, Functions, Series, Operators, Budapest, 1980, Colloquia Mathematica Societatis Jànos Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 509-524.
17. Feichtinger H.G., and Weisz F. Gabor analysis on Wiener amalgams. Sampl. Theory Signal Image Process. 6 2 (2007) 129-150
18. Folland G.B. Harmonic Analysis in Phase Space (1990), Princeton University Press, Princeton, NJ
19. Gohberg I., Kaashoek M.A., and Woerderman H.J. The band method for positive and strictly contractive extension problems: an alternative version and new applications. Integral Equations Operator Theory 12 3 (1989) 343-382
20. Gröchenig K. Foundations of Time-Frequency Analysis (2001), Birkhäuser, Boston
21. K. Gröchenig, Weight functions in time-frequency analysis, in: L. Rodino et al. (Eds.), Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Comm., Vol. 52, pp. 343-366, 2007.
22. Gröchenig K., Heil C., and Okoudjou K. Gabor analysis in weighted amalgam spaces. Sampl. Theory Signal Image Process. 1 3 (2002) 225-259
23. Gröchenig K., and Leinert M. Wiener's lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc. 17 1 (2003) 1-18
24. Heil C. An introduction to weighted Wiener amalgams. In: Krishna M., Radha R., and Thangavelu S. (Eds). Wavelets and their Applications, Chennai, January 2002 (2003), Allied Publishers, New Delhi 183-216
25. Heil C., and Walnut D. Continuous and discrete wavelet transforms. SIAM Rev. 31 (1989) 628-666
26. Jaffard S. Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 5 (1990) 461-476
27. Janssen A.J.E.M. Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1 4 (1995) 403-436
28. Kurbatov V. Some algebras of operators majorized by convolution. Funct. Differential Equations 8 3-4 (2001) 323-333 International Conference on Differential and Functional Differential Equations, Moscow, 1999
29. Kurbatov V.G. Algebras of difference and integral operators. Funct. Anal. Appl. 24 2 (1990) 156-158
30. Kurbatov V.G. Functional Differential Operators and Equations, Mathematics and Its Applications vol. 473 (1999), Kluwer Academic Publishers, Dordrecht, Boston, London
31. d). Duke Math. J. 89 (1997) 237-282
32. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, 21 pp., École Polytech., Palaiseau, 1995.
33. M.A. Šubin [Shubin], Almost periodic functions and partial differential operators, (Russian) Uspehi Mat. Nauk 33 (2) (1978) 3-47 (translation in Russian Math. Surveys 33 (2) (1978) 1-52).
34. Sun Q. Wiener's lemma for infinite matrices with polynomial off-diagonal decay. C. R. Math. Acad. Sci. Paris 340 8 (2005) 567-570
35. Walnut D.F. Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165 2 (1992) 479-504
36. Wexler J., and Raz S. Discrete Gabor expansions. Signal Process. 21 (1990) 207-221
37. Wiener N. Tauberian theorems. Ann. of Math. 33 (1932) 1-100