Time-frequency analysis associated with the generalized Gabor transform and applications

Abstract

In this article, we introduce and study the generalized Gabor transform associated with a class of operators of Sturm-Liouville. We investigate for this transform some problems of time-frequency analysis. In particular, we study the concept of the localization operators and the spectrogram analysis linked to the new generalized Gabor transform. We examine a particular class of the localization operators known as concentration operators. We prove that a finite vector space generated by the first eigenfunctions of such operators has a maximal time-frequency concentration within the region of interest. Then we will use it to approximate functions which are almost concentrated in such a region.

Keywords

• Chébli-Trimèche hypergroup
• generalized linear canonical Fourier transform
• generalized Gabor transform
• localization operators

References

1. [1] L.D. Abreu, K. Gröchenig and J.L. Romero, On accumulated spectrograms, Trans. Amer. Math. Soc. 368, 3629–3649, 2016.
2. [2] A. Achour and K. Trimèche, La g-fonction de Littlewood-Paley associée à un opérateur différentiel singulier sur $(0,\infty)$, Ann. Inst. Fourier Grenoble 33, 203-226, 1983.
3. [3] N. Andersen, On the Range of the Chébli-Trimèche Transform, Mh Math 144, 193-201, 2005.
4. [4] L.B. Attour and K. Trimèche, Uncertainty principle and $(L^{ p}, L^{q})$ sufficient pairs on Chébli- Trimèche hypergroups, Integral Transforms Spec. Funct. 16 (8), 625-637, 2005.
5. [5] P. Balazs, Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolution Inf. Process 6, 315–330, 2008.
6. [6] W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, Volume 20 of de Gruyter Studies in Mathematics, Walter de Gruyter Co., Berlin, 1995.
7. [7] W. R. Bloom and Z. F. Xu, The Hardy-Littlewood maximal function for Chébli-Trimèche hypergroups, 4570 in Applications of hypergroups and related measure algebras (Seattle, WA, 1993), edited by W. C. Connett et al., Contemp. Math. 183, Amer. Math. Soc., Providence, RI, 1995.
8. [8] W. R. Bloom and Z. F. Xu, Fourier transforms of Schwartz functions on Chébli-Trimèche hypergroups, Monatsh. Math. 125, 89-109, 1998.

9. [9] P. Boggiatto and M.W. Wong, Two-wavelet localization operators on $L^{p}(\mathbb{R}^{n})$ for the Weyl- Heisenberg group, Integral Equ. Oper. Theory, 49, 1–10, 2004.
10. [10] F. Bouzeffour, M. Dziri and A. Fitouhi, Variation diminishing convolution kernels associated with second-order differential operators, Integ. Trans. Spec. Functions 24 (12), 1012-1026, 2013.
11. [11] P. Bremaud, Mathematical Principles of Signal Processing, Fourier and Wavelet Analysis, Springer, New York, 2002.
12. [12] H. Chébli, Sur un théorème de Paley-Wiener associé à la décomposition spectrale d’un opérateur de Sturm-Liouville sur $(0,\infty)$, J. Funct. Anal. 17, 447461, 1974.
13. [13] H. Chébli, Opêrateur de convolution généralisée et semi groupe de convolution, Lecture Notes in Mathematics Théorie du Potentiel et Analyse Harmonique, 404, 35-59, Springer- Verlag Berlin Heidelberg, 1974.
14. [14] H. Chébli, Sturm-Liouville hypergroups, In Applications of hypergroups and related measure algebras. (Seattle, WA, 1993), 183 of Contemp. Math. 71-88. Amer. Math. Soc., Providence, RI. 1995.
15. [15] E. Cordero and K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205, 107–131, 2003.
16. [16] W. Czaja and G. Gigante, Continuous Gabor transform for strong hypergroups, J Fourier Anal Appl. 9, 321-339, 2003.
17. [17] I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory, 34, 605–612, 1988.
18. [18] F. De Mari H. Feichtinger and K. Nowak, Uniform eigenvalue estimates for time-frequency localization operators, J. London Math. Soc. 65, 720–732, 2002.
19. [19] A. G. Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups, Bull. Malaysian Math. Soc. 38, 779-803, 2015.
20. [20] A. G. Farashahi, R. Kamyabi-Gol, Continuous Gabor transform for a class of non-Abelian groups, Bull. Belg. Math. Soc. 19 (4), 683-701, 2012.
21. [21] H. G. Feichtinger, W. Kozek and F. Luef, Gabor analysis over finite abelian groups, Appl. Comput. Harmon. Anal. 26 (2), 230-248, 2009.
22. [22] H. G. Feichtinger and T. Strohmer, Advances in Gabor Analysis, Birkhäuser, Boston, 2003.
23. [23] A. Fitouhi, A. Heat polynomials for a singular differential operator on $(0,\infty)$, Constr. Approx 5, 241-270, 1989.
24. [24] D. Gabor, Theory of communication. part 1: The analysis of information, Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, 93 (26), 429-441, 1946.
25. [25] S. Ghobber, Dunkl-Gabor transform and time-frequency concentration, Czechoslov. Math. J. 65 (1), 255-270, 2015.
26. [26] S. Ghobber and S. Omri, Time-frequency concentration of the windowed Hankel transform, Integ. Transf. Spec. Funct. 25 (6), 481-496, 2014.
27. [27] K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and algorithms, 211-231, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998.
28. [28] K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
29. [29] L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328, 1484–1486, 2007.
30. [30] R. Ma, Heisenberg uncertainty principle on Chébli-Trimèche hypergroups, Pacific J. Math. 235 (2), 289-296, 2008.
31. [31] H. Mejjaoli, k-Hankel Gabor transform on $\mathbb{R}^{d}$ and its applications to the reproducing kernel theory, Complex Anal. Oper. Theory 15 (14), 1-54, 2021.
32. [32] H. Mejjaoli, Jacobi-Dunkl linear canonical windowed transform and applications to the reproducing kernel theory and time-frequency analysis, In Revision.
33. [33] H. Mejjaoli, Time-frequency analysis and quantitative uncertainty principles associated with the Jacobi-Cherednik linear canonical Gabor transform, Preprint 2025.
34. [34] H. Mejjaoli and A. Poria, The linear canonical transforms associated with the Dunkl-type operator and its applications, Integral Transforms Spec. Funct. 1-40, 2025.
35. [35] H. Mejjaoli and N. Sraieb, Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform. Mediterr. J. Math., 5, 443-466, 2008.
36. [36] H. Mejjaoli and N. Sraieb, Gabor transform in quantum calculus and applications, Fractional Calculus and Applied Analysis, 12 (3), 319-336, 2009.
37. [37] F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publishing Co., New York, 1995.
38. [38] N. Sraeib, Uncertainty principles for the q-Bessel windowed transform and localization operators, Journal of Mathematical Sciences, 271, 434-457, 2023.
39. [39] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83, 482–492, 1956.
40. [40] K. Trimèche, Convergence des series de Taylor generalisées au sens de Delsarte, C. R. Acad. Sci. Paris Sér. A, 281, 1015-1017, 1975.
41. [41] K. Trimèche, Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur $(0,\infty)$, J. Math. Pures Appl. 60, 51-98, 1981.
42. [42] K. Trimèche, Inversion of the Lions transmutation operators using generalized wavelets, App. Comput. Harm. Anal. 4 97-112, 1997.
43. [43] K. Trimèche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, 1997.
44. [44] K. Trimèche, Cowling-Price and Hardy theorems on Chébli-Trimèche hypergroups, Glob. J. Pure Appl. Math. 1 (3), 286305, 2005.
45. [45] E. Wilczok, New uncertainty principles for the continuous Gabor transform and the continuous Gabor transform, Doc. Math., J. DMV (electronic) 5, 201-226, 2000.
46. [46] M. W. Wong, Localization operators on the Weyl-Heisenberg group, in: Geometry, Analysis and Applications. Editor: R.S. Pathak, World-Scientific; 303-314, 2001.
47. [47] M. W. Wong, $L^{p}$ boundedness of localization operators associated to left regular representations, Proc. Amer. Math. Soc. 130, 2911-2919, 2002.
48. [48] M. W. Wong, Localization operators on the affine group and paracommutators, in Progress in Analysis, World Scientific, 663-669, 2003.
49. [49] M. W. Wong, Wavelet transforms and localization operators, 136, Springer Science & Business Media, 2002