Two-dimensional fractional Fourier transform and some of its properties. Issue 7 (3rd July 2018)

• Record Type: Journal Article
• Authors: Zayed, Ahmed
• Is Part Of: Integral transforms and special functions. Volume 29:Issue 7(2018)
• Page Start: 553
• Page End: 570
• Publication Date: 2018-07-03
• Subjects: Fractional Fourier transform -- two-dimensional fractional Fourier transform -- Hermite polynomials of two complex variables -- convolution theorem -- Poisson summation formula; Primary: 42B10 -- 42C05 -- Secondary: 33C50 -- 94A11; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2018.1471689
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. It is known that the Hermite functions of order n are eigenfunctions of the Fourier transform with eigenvalues( i ) n and likewise the Hermite functions of order n are eigenfunctions of the FrFT of order α, where0 ≤ α ≤ 1 but with different eigenvaluesλ n . Those eigenvaluesλ n approach( i ) n as α approaches 1. The FrFT has been extended to n dimensions using tensor product of n copies of the one-dimensional transform. In this article we introduce a new two-dimensional FrFT that is not a tensor product of two one-dimensional transforms. The definition of the new transform is based on using the Hermite functions of two complex variables as eigenfunctions of the transform. We then derive some of its properties, such as its inversion formula, convolution structure and theorem, and its analogue of Poisson summation formula.