Abstract
The main objective of this paper is to study and give basic properties of the fractional Fourier–Jacobi type transform of rapidly decreasing functions and of tempered distributions. An application in solving a generalized heat equation is given.
Similar content being viewed by others
References
Almeida, L.B.: The fractional Fourier transform and time–frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994)
Chaudhary, M.S., Thorat, S.P.: On Fourier–Jacobi transformation. Indian J. Pure Appl. Math 27(2), 157–164 (1996)
Flensted-Jensen, M.: Paley–Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. 10, 143–162 (1972)
Flensted-Jensen, M., Koornwinder, T.H.: Jacobi functions: the addition formula and the positivity of dual convolution structure. Ark. Mat. 17, 139–151 (1979)
Flensted-Jensen, M., Koornwinder, T.H.: The convolution structure for Jacobi function expansions. Ark. Mat. 11, 245–262 (1973)
Kawazoe, T., Liu, J.: Fractional calculus and analytic continuation of the complex Fourier–Jacobi transform. Tokyo J. Math. 27(1), 187–207 (2004)
Koornwinder, T.H.: A new proof of a Paley–Wiener type theorem for the Jacobi transform. Ark. Mat. 13, 145–159 (1975)
Mizony, M.: Une transformation de Laplace–Jacobi. SIAM J. Math. Anal. 14(5), 987–1003 (1983)
Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25, 241–265 (1980)
Pathak, R.S., Prasad, A., Kumar, M.: Fractional Fourier transform of tempered distributions and generalized pseudo-differential operator. J. Pseudo-differ. Oper. Appl. 3, 239–254 (2012)
Prasad, A., Singh, V.K.: The fractional Hankel transform of certain tempered distributions and pseudo-differential operators. Ann. Univ. Ferrara 59(1), 141–158 (2013)
Trimèche, K.: Inversion of the Lions transmutation operators using generalized wavelets. Appl. Comput. Harmon. Anal. 4(1), 97–112 (1997)
Zayed, A.I.: On the relationship between the Fourier and fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310–311 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sahbani, S. Fractional Fourier–Jacobi type transform. Ann Univ Ferrara 66, 135–156 (2020). https://doi.org/10.1007/s11565-020-00337-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-020-00337-3
Keywords
- Fourier–Jacobi transform
- Fractional Fourier–Jacobi type transform
- Tempered distribution
- Generalized heat equation