Advertisement

Fractional Fourier transformation of Schwartz test functions

  • Saleh AbdullahEmail author
  • Bushra Elshatarat
Article

Abstract

In this paper we study the fractional Fourier transformation on the space S of Schwartz test functions, study some of its properties, and establish a two sided inverse for it. Also, we establish a convolution theorem for the fractional Fourier transform. We use duality to define fractional Fourier transform of tempered distributions. We define fractional convolution of a function and a tempered distribution and fractional convolution of tempered distributions, and show continuity of the convolution operators involved.

Keywords

Fractional Fourier transform Schwartz test functions Tempered distribution 

Mathematics Subject Classification

46F12 42A38 

Notes

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interests.

References

  1. 1.
    Barros-Netto, J.: An Introduction to the Theory of Distributions. Marcel Dekker, New York (1973)Google Scholar
  2. 2.
    Kerr, F.H.: A Distributional Approach to Namias’ Fractional Fourier Transform. In: Proc. of the Royal Soc. of Edinburgh, 188A, pp. 133–143 (1988)Google Scholar
  3. 3.
    Khan, K.N., Lamb, W., McBride, A.: Fractional transform of generalized functions. Integral Transform. Special Funct. 20(61), 471–490 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Kilbas, A.A., Luchko, YuF, Matrines, H., Trujillo, J.: Fractional Fourier transform in the framework of fractional operators. Integral Transform. Special Funct. 21(10), 779–795 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Luchko, Yu, F., Matrines, H., Trujillo, J.: Fractional Fourier transform and some of its applications. Fract. Calc. Appl. Anal. 11(4), 457–470 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    McBride, A., Kerr, F.: On Namias’ fractional Fourier transforms. IMA J. Appl. Math. 39, 159–175 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)zbMATHGoogle Scholar
  9. 9.
    Zayed, A.I.: Fractional Fourier transform of generalized functions. Integral Transform. Special Funct. 7(3–4), 299–312 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zayed, A.I.: A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)CrossRefGoogle Scholar

Copyright information

© Unione Matematica Italiana 2019

Authors and Affiliations

  1. 1.Department of MathematicsJordan University of Science And TechnologyIrbidJordan

Personalised recommendations