CHARACTERIZATION OF RATIONAL TIME-FREQUENCY MULTI-WINDOW GABOR FRAMES AND THEIR DUALS
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
This paper addresses multi-window Gabor frames with ratio- nal time-frequency product. Such issue was considered by Zibulski and Zeevi (Appl. Comput. Harmonic Anal. 4 (1997), 188–221) in terms of Zak transform matrix (so-called Zibuski-Zeevi matrix), and by many oth- ers. In this paper, we introduce of a new Zak transform matrix. It is different from Zibulski-Zeevi matrix, but more direct and convenient for our purpose. Using such Zak transform matrix we characterize rational time-frequency multi-window Gabor frames (Riesz bases and orthonor- mal bases), and Gabor duals for a Gabor frame. Some examples are also provided, which show that our Zak transform matrix method is efficient.
Abstract
This paper addresses multi-window Gabor frames with ratio- nal time-frequency product. Such issue was considered by Zibulski and Zeevi (Appl. Comput. Harmonic Anal. 4 (1997), 188–221) in terms of Zak transform matrix (so-called Zibuski-Zeevi matrix), and by many oth- ers. In this paper, we introduce of a new Zak transform matrix. It is different from Zibulski-Zeevi matrix, but more direct and convenient for our purpose. Using such Zak transform matrix we characterize rational time-frequency multi-window Gabor frames (Riesz bases and orthonor- mal bases), and Gabor duals for a Gabor frame. Some examples are also provided, which show that our Zak transform matrix method is efficient.
참고문헌 (21)
- M. A. Akinlar, "Oblique duals associated with rational subspace Gabor frames", J. Integral Equations Appl, 20(3), pp.283~309 (2008)
- I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis", IEEE Trans. Inform. Theory, 36(5), pp.961~1005 (1990)
- H. G. Feichtinger, "Gabor Analysis and Algorithms, Theory and Applications" (1998), Birkhauser
- H. G. Feichtinger, "Advances in Gabor Analysis" (2003), Birkhauser
- J.-P. Gabardo, "Balian-Low phenomenon for subspace Gabor frames", J. Math. Phys, 45(8), pp.3362~3378 (2004)
- J.-P. Gabardo, "The uniqueness of the dual of Weyl-Heisenberg subspace frames", Appl. Comput. Harmon. Anal, 17(2), pp.226~240 (2004)
- J.-P. Gabardo, "Density results for Gabor systems associated with periodic subsets of the real line", J. Approx. Theory, 157(2), pp.172~192 (2009)
- K. Grochenig, "Foundations of Time-Frequency Analysis" (2001), Birkhauser
- C. Heil, "History and evolution of the density theorem for Gabor frames", J. Fourier Anal. Appl, 13(2), pp.113~166 (2007)
- F. Jaillet, "Time-frequency jigsaw puzzle: adaptive multiwindow and multilayered Gabor expansions", Int. J. Wavelets Multiresolut. Inf. Process, 5(2), pp.293~315 (2007)
- S. Jang, "Compactly supported multiwindow dual Gabor frames of rational sampling density", Adv. Comput. Math, 38(1), pp.159~186 (2013)
- S. Jang, "Techniques for smoothing and splitting in the construction of tight frame Gabor windows", Int. J. Wavelets Multiresolut. Inf. Process, 11(1), pp.18~ (2013)
- S. Li, "Discrete multi-Gabor expansions", IEEE Trans. Inform. Theory, 45(6), pp.1954~1967 (1999)
- Y.-Z. Li, "Gabor systems on discrete periodic sets", Sci. China Ser. A, 52(8), pp.1639~1660 (2009)
- Q.-F. Lian, "The duals of Gabor frames on discrete periodic sets", J. Math. Phys, 50(1), pp.22~ (2009)
- Q.-F. Lian, "Gabor frame sets for subspaces", Adv. Comput. Math, 34(4), pp.391~411 (2011)
- A. Ron, "Affine systems in L2(Rd) : the analysis of the analysis operator", J. Funct. Anal, 148(2), pp.408~447 (1997)
- R. M. Young, "An Introduction to Nonharmonic Fourier Series" (1980), Academic Press
- Y. Y. Zeevi, "Twentieth century harmonic analysis-a celebration", pp.173~199 (2001), Kluwer Acad. Publ.
- Y. Y. Zeevi, "Gabor analysis and algorithms", pp.381~407 (1998), Birkhauser Boston
- M. Zibulski, "Analysis of multiwindow Gabor-type schemes by frame methods", Appl. Comput. Harmon. Anal, 4(2), pp.188~221 (1997)