On the class of uncertainty inequalities for the coupled fractional Fourier transform
Firdous A. Shah, Waseem Z. Lone, Kottakkaran Sooppy Nisar, Thabet Abdeljawad
Abstract
Abstract The coupled fractional Fourier transform $\mathcal {F}_{\alpha ,\beta}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:math> 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 $\gamma =(\alpha +\beta )/2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math> and $\delta =(\alpha -\beta )/2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>δ</mml:mi><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math> . 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.