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

The coupled fractional Fourier transform F α , β 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 γ = ( α + β ) / 2 and 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.