Fractional Fourier transform on R<sup>2</sup> and an application

Yue ZHANG; Wenjuan LI

doi:10.1007/s11464-021-0983-9

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Front. Math. China ›› 2022, Vol. 17 ›› Issue (6) : 1181-1200. DOI: 10.1007/s11464-021-0983-9

RESEARCH ARTICLE

Fractional Fourier transform on R² and an application

Yue ZHANG ,
Wenjuan LI

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Abstract

We focus on the L^p(R²) theory of the fractional Fourier transform (FRFT) for 1 ≤ p ≤ 2. In L¹(R²), we mainly study the properties of the FRFT via introducing the two-parameter chirp operator. In order to get the point-wise convergence for the inverse FRFT, we introduce the fractional convolution and establish the corresponding approximate identities. Then the well-defined inverse FRFT is given via approximation by suitable means, such as fractional Gauss means and Able means. Furthermore, if the signal F_α,βf is received, we give the process of recovering the original signal f with MATLAB. In L²(R²), the general Plancherel theorem, direct sum decomposition, and the general Heisenberg inequality for the FRFT are obtained.

Keywords

Fractional Fourier transform (FRFT) / inverse fractional Fourier transform / signal recovery / direct sum decomposition / general Heisenberg inequality

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Yue ZHANG, Wenjuan LI. Fractional Fourier transform on R² and an application. Front. Math. China, 2022, 17(6): 1181‒1200 https://doi.org/10.1007/s11464-021-0983-9

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