Wireless Personal Communications

, Volume 92, Issue 2, pp 623–637 | Cite as

Two New Convolutions for the Fractional Fourier Transform

  • P. K. Anh
  • L. P. Castro
  • P. T. Thao
  • N. M. Tuan
Article

Abstract

In this paper we introduce two novel convolutions for the fractional Fourier transforms, and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity, which may be useful in signal processing and other types of applications. We analyze a consequent comparison with other known convolutions, and establish necessary and sufficient conditions for the solvability of associated convolution equations of both the first and second kind in \(L^1({\mathbb {R}})\) and \(L^2 ({\mathbb {R}})\) spaces. An example satisfying the sufficient and necessary condition for the solvability of the equations is given at the end of the paper .

Keywords

Convolution Convolution theorem Fractional Fourier transform Convolution equation Filtering 

Mathematics Subject Classification

40E99 43A32 47B15 44A20 68T37 94A12 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • P. K. Anh
    • 1
  • L. P. Castro
    • 2
  • P. T. Thao
    • 3
  • N. M. Tuan
    • 4
  1. 1.Department of Computational and Applied Mathematics, College of ScienceVietnam National UniversityHanoiViet Nam
  2. 2.Department of Mathematics, Center for R&D in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  3. 3.Department of MathematicsHanoi Architectural UniversityHanoiViet Nam
  4. 4.Department of Mathematics, College of EducationVietnam National UniversityHanoiViet Nam

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