International Journal of Nonlinear Analysis and Applications
Some inequalities for linear canonical curvelet transform
Document Type : Research Paper
Authors
1 Engineering Mathematics Section, Faculty of Engineering and Technology, Annamalai University Annamalai Nagar, 608002, Tamil Nadu, India
2 Department of Mathematics, Annamalai University, Annamalai Nagar, 608002, Tamil Nadu, India
Abstract
In this paper, we derive some inequalities for linear canonical curvelet transform (LCCT). At the outset, the basic properties of LCCT including the admissibility condition, and Moyal's principle are stated. Thereafter, some notable inequalities and results related to the well-known Heisenberg- type inequalities are derived for linear canonical curvelet transform.
Keywords
[1] L.B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process. 42 (1994), 3084–3091.
[2] M. Bahri and R.Ashino, Some properties of windowed linear canonical transform and its logarithmic uncertainty principle, Int. J. Wavelets Multiresol. Inf. Process. 14 (2016), 1650015.
[3] B. Barshan, M.A. Kutay and H.M. Ozaktas, Optimal filtering with linear canonical transformations, Opt. Commun. 135 (1997), 32–36.
[4] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897–1905.
[5] R. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill Book Co., New York, 1986.
[6] E.J. Cand´es and D.L. Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal. 19 (2005), 162–197.
[7] E.J. Cand´es and D.L. Donoho, Continuous curvelet transform: II. Discretization and frames, Appl. Comput. Harmon. Anal. 19 (2005), 198–222.
[8] E.J. Cand´es and D.L. Donoho, Ridgelets: a key to higher dimensional intermittency, Phil. Trans. Royal Soc. Lond. Ser. A: Math. Phys. Engin. Sci. 357 (1999), 2495–2509.
[9] S.A. Collins, Lens-system diffraction integral written in terms of matrix optics, J. Optic. Soc. Amer. 60 (1970), 1168–1177.
[10] M.G. Cowling and J.F. Price , Bandwidth verses time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math Anal. 15 (1984), 151–165.
[11] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106.
[12] X. Guanlei, W. Xiaotong and X. Xiaogang, Three uncertainty relations for real signals associated with linear canonical transform, IET Signal Process. 3 (2009), 85–92.
[13] J.J. Healy, M.A. Kutay, H.M. Ozaktas and J.T. Sheridan, Linear Canonical Transforms, Springer, New York 2016.
[14] J.J. Healy and J.T. Sheridan, Sampling and discretization of the linear canonical transform, Signal Process. 89 (2009), 641–648.
[15] H. Huo, Uncertainty Principles for the Offset Linear Canonical Transform, Circuits Syst. Signal Process. 38 (2019), 395–406.
[16] D.F.V. James and G. S. Agarwal, The generalized Fresnel transform and its applications to optics, Opt. Commun. 126 (1996), 207–212.
[17] P. Jaming, Nazarov’s uncertainty principles in higher dimension, J. Approx. Theory 149 (2007) 30–41.
[18] A.A. Khan and K. Ravikumar, Linear canonical curvelet transform and the associated Heisenberg-type inequalities, Int. J. Geomet. Meth. Mod. Phys. 18 (2021), no. 7.
[19] G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc. 361 (2009), 2719–2754.
[20] J. Li and M.W. Wong, Localization operators for curvelet transforms, J. Pseudo-Differ. Oper. Appl. 3 (2012), 121–143.
[21] M. Moshinsky and C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971), 1772–1780.
[22] D. Mustard, Uncertainty principle invariant under fractional Fourier Transform, J. Aust. Math. Soc. Ser. B 33 (1991), 180–191.
[23] F.L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra Anal. 5 (1993), 3–66.
[2] M. Bahri and R.Ashino, Some properties of windowed linear canonical transform and its logarithmic uncertainty principle, Int. J. Wavelets Multiresol. Inf. Process. 14 (2016), 1650015.
[3] B. Barshan, M.A. Kutay and H.M. Ozaktas, Optimal filtering with linear canonical transformations, Opt. Commun. 135 (1997), 32–36.
[4] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897–1905.
[5] R. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill Book Co., New York, 1986.
[6] E.J. Cand´es and D.L. Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal. 19 (2005), 162–197.
[7] E.J. Cand´es and D.L. Donoho, Continuous curvelet transform: II. Discretization and frames, Appl. Comput. Harmon. Anal. 19 (2005), 198–222.
[8] E.J. Cand´es and D.L. Donoho, Ridgelets: a key to higher dimensional intermittency, Phil. Trans. Royal Soc. Lond. Ser. A: Math. Phys. Engin. Sci. 357 (1999), 2495–2509.
[9] S.A. Collins, Lens-system diffraction integral written in terms of matrix optics, J. Optic. Soc. Amer. 60 (1970), 1168–1177.
[10] M.G. Cowling and J.F. Price , Bandwidth verses time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math Anal. 15 (1984), 151–165.
[11] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106.
[12] X. Guanlei, W. Xiaotong and X. Xiaogang, Three uncertainty relations for real signals associated with linear canonical transform, IET Signal Process. 3 (2009), 85–92.
[13] J.J. Healy, M.A. Kutay, H.M. Ozaktas and J.T. Sheridan, Linear Canonical Transforms, Springer, New York 2016.
[14] J.J. Healy and J.T. Sheridan, Sampling and discretization of the linear canonical transform, Signal Process. 89 (2009), 641–648.
[15] H. Huo, Uncertainty Principles for the Offset Linear Canonical Transform, Circuits Syst. Signal Process. 38 (2019), 395–406.
[16] D.F.V. James and G. S. Agarwal, The generalized Fresnel transform and its applications to optics, Opt. Commun. 126 (1996), 207–212.
[17] P. Jaming, Nazarov’s uncertainty principles in higher dimension, J. Approx. Theory 149 (2007) 30–41.
[18] A.A. Khan and K. Ravikumar, Linear canonical curvelet transform and the associated Heisenberg-type inequalities, Int. J. Geomet. Meth. Mod. Phys. 18 (2021), no. 7.
[19] G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc. 361 (2009), 2719–2754.
[20] J. Li and M.W. Wong, Localization operators for curvelet transforms, J. Pseudo-Differ. Oper. Appl. 3 (2012), 121–143.
[21] M. Moshinsky and C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971), 1772–1780.
[22] D. Mustard, Uncertainty principle invariant under fractional Fourier Transform, J. Aust. Math. Soc. Ser. B 33 (1991), 180–191.
[23] F.L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra Anal. 5 (1993), 3–66.
[24] H.M. Ozaktas, Z. Zalevsky and M.A. Kutay, The fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, New York, 2000.
[25] S. C. Pei and J. J. Ding, Eigen functions of the offset Fourier, fractional Fourier and linear canonical transform, J. Opt. Soc. Amer. 20 (2003), 522–532.
[26] F.A. Shah and A.Y. Tantary, Non-isotropic angular Stockwell transform and the associated uncertainty principles, Appl. Anal. 100 (2021), 835–859.
[27] K.K. Sharma and S.D. Joshi, Uncertainty principles for real signals in linear canonical transform domains, IEEE Trans. Signal Process. 56 (2008) 2677–2683.
[28] J. Shi, X. Liu and N. Zhang, Generalized convolution and product theorems associated with linear canonical transform, SIViP 8 (2014), 967–974.
[29] A. Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Am A. 25 (2008), 647–652.
[30] T.Z. Xu and B.Z. Li, Linear Canonical Transform and Its Applications, Science Press, Beijing, China 2013.
[31] A. I. Zayed, Sampling of signals bandlimited to a disc in the linear canonical transform domain, IEEE Signal Process. Lett. 25 (2018), 1765–1769.
[32] Q. Zhang, Zak transform and uncertainty principles associated with linear canonical transform, IET Signal Process. 10 (2016), 791–797.
[33] J. Zhao, R. Tao, Y. Li and Y. Wang, Uncertainty principles for linear canonical transform, IEEE Trans, Signal Process. 57 (2009), 2856–2858.
[25] S. C. Pei and J. J. Ding, Eigen functions of the offset Fourier, fractional Fourier and linear canonical transform, J. Opt. Soc. Amer. 20 (2003), 522–532.
[26] F.A. Shah and A.Y. Tantary, Non-isotropic angular Stockwell transform and the associated uncertainty principles, Appl. Anal. 100 (2021), 835–859.
[27] K.K. Sharma and S.D. Joshi, Uncertainty principles for real signals in linear canonical transform domains, IEEE Trans. Signal Process. 56 (2008) 2677–2683.
[28] J. Shi, X. Liu and N. Zhang, Generalized convolution and product theorems associated with linear canonical transform, SIViP 8 (2014), 967–974.
[29] A. Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Am A. 25 (2008), 647–652.
[30] T.Z. Xu and B.Z. Li, Linear Canonical Transform and Its Applications, Science Press, Beijing, China 2013.
[31] A. I. Zayed, Sampling of signals bandlimited to a disc in the linear canonical transform domain, IEEE Signal Process. Lett. 25 (2018), 1765–1769.
[32] Q. Zhang, Zak transform and uncertainty principles associated with linear canonical transform, IET Signal Process. 10 (2016), 791–797.
[33] J. Zhao, R. Tao, Y. Li and Y. Wang, Uncertainty principles for linear canonical transform, IEEE Trans, Signal Process. 57 (2009), 2856–2858.
)
Volume 14, Issue 1
January 2023Pages 2361-2372
- Receive Date: 27 September 2021
- Revise Date: 04 October 2021
- Accept Date: 05 December 2022