TY - JOUR T1 - Pseudo-differential operators associated with the gyrator transform AU - Mahato, Kanailal AU - Arya, Shubhanshu AU - Prasad, Akhilesh PY - 2025 DA - August Y2 - 2024 DO - 10.15672/hujms.1471348 JF - Hacettepe Journal of Mathematics and Statistics PB - Hacettepe University WT - DergiPark SN - 2651-477X SP - 1426 EP - 1441 VL - 54 IS - 4 LA - en AB - In this present work, a brief introduction to the gyrator transform and its fundamental properties are given. {The gyrator transform of tempered distributions is being discussed}. This article made further discussion on the boundedness properties of pseudo-differential operators associated with the gyrator transform on Schwartz space as well as on Sobolev space. KW - Fourier transform KW - Fractional Fourier transform KW - Gyrator transform KW - tempered distribution KW - pseudo-differential operator CR - [1] L.B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process. 42, 3084-3091, 1994. CR - [2] H. Dai, Z. Zheng and W. Wang, A new fractional wavelet transform, Commun. Nonlinear Sci. Numer. Simul. 44, 19-36, 2017. CR - [3] A. Friedman, Generalized functions and partial differential equations, Englewood Cliffs NJ, Prentice Hall, 1963. CR - [4] L. Hörmander, The Analysis of Linear Partial Differential Operators, (I-III), Berlin, Springer, (1983-1985), 2007. CR - [5] T. Kagawa and T. Suzuki, Characterizations of the gyrator transform via the fractional Fourier transform, Integral Transforms Spec. Funct. 34 (5), 399-413, 2023. CR - [6] K. Mahato and S. Arya, Gyrator potential operator and Lp−Sobolev spaces involving Gyrator transform, Integral Transforms Spec. Funct. Accepted, 2024. CR - [7] D. Mendlovic and H.M. Ozaktas, Fractional Fourier transforms and their optical implementation I, J. Opt. Soc. Amer. A. 10, 1875-1881, 1993. CR - [8] V. Namias, The fractional order Fourier transform and its applications to quantum mechanics, IMA J. Appl. Math. 25, 241-261, 1980. CR - [9] H.M. Ozaktas, Z. Zalevsky and M. Kutay, The fractional Fourier transform with applications in optics and signal processing, New York, John Wiley, 2001. CR - [10] R.S. Pathak, A. Prasad and M. Kumar, Fractional Fourier transform of tempered distributions and generalized Pseudo-differential Operators, J. Pseudo-Diff. Oper. Appl. 3 (2), 239-254, 2012. CR - [11] A. Prasad and M. Kumar, Product of two generalized Pseudo-differential Operators involving fractional Fourier transform, J. Pseudo-Diff. Oper. Appl. 2 (3), 355-365, 2011. CR - [12] A. Prasad and M. Kumar, Boundedness of Pseudo-differential operator associated with fractional Fourier transform, Proc. Natl. Acad. Sci. India Sect. A .Phys. Sci. 84 (4), 549-554, 2014. CR - [13] A. Prasad and P. Kumar, Pseudo-differential operator associated with the fractional Fourier transform, Math. Commun. 21, 115-126, 2016. CR - [14] A. Prasad, S. Manna and A. Mahato, The generalized continuous wavelet transform associated with the fractional Fourier transform, J. Comput. Appl. Math. 259, 660- 671, 2014. CR - [15] L. Rodino, Linear Partial Differential Operators in Gevrey spaces, Singapore, World Scientific, 1993. CR - [16] J.A. Rodrigo, T. Alieva and M.L. Calvo, Gyrator transform: properties and applications, Opt. Express. 15 (5), 2190-2203, 2007. CR - [17] R. Simon and K.B. Wolf, Structure of the set of paraxial optical systems, J. Opt. Soc. Am. 17 (2), 342-355, 2000. CR - [18] M.W. Wong, An Introduction to Pseudo-differential Operators, 2nd edn. Singapore, World Scientific, 1999. CR - [19] S. Zaidman, Distributions and Pseudo-differential operators, Longman, Essex, 1991. UR - https://doi.org/10.15672/hujms.1471348 L1 - https://dergipark.org.tr/en/download/article-file/3875587 ER -