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

Abstract

AbstractThe coupled fractional Fourier transform$\mathcal {F}_{\alpha ,\beta}$Fα,βis a two-dimensional fractional Fourier transform depending on two anglesαandβ, which are coupled in such a way that the transform parameters are$\gamma =(\alpha +\beta )/2$γ=(α+β)/2and$\delta =(\alpha -\beta )/2$δ=(α−β)/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.