On the class of uncertainty inequalities for the coupled fractional Fourier transform

Abstract

The coupled fractional Fourier transform mathcal {F}_{alpha ,beta} is a two-dimensional fractional Fourier transform depending on two angles α and β, which are coupled in such a way that the transform parameters are gamma =(alpha +beta )/2 and delta =(alpha -beta )/2. It generalizes the two-dimensional Fourier transform and serves as a prominent tool in some applications of signal and image processing. In this article, we formulate a new class of uncertainty inequalities for the coupled fractional Fourier transform (CFrFT). Firstly, we establish a sharp Heisenberg-type uncertainty inequality for the CFrFT and then formulate some logarithmic and local-type uncertainty inequalities. In the sequel, we establish several concentration-based uncertainty inequalities, including Nazarov, Amrein–Berthier–Benedicks, and Donoho–Stark’s inequalities. Towards the end, we formulate Hardy’s and Beurling’s inequalities for the CFrFT.