Uncertainty Principles for Rényi Entropy Under Continuous and Discrete Quaternion Fourier Transforms

We first prove the sharp Hausdorff–Young inequality for the right‐sided continuous quaternion Fourier transform (RCQFT), and the Hausdorff–Young inequalities for the discrete quaternion Fourier transform (DQFT) and for the quaternion Fourier series (QFS), respectively. Next, on the basis of the measure of uncertainty for the Rényi entropy in the quantum world, the uncertainty principle (UP) of the Rényi entropy is established for position and momentum. Moreover, similar uncertainty relations are also derived for the ‐level systems and for the angle and angular momentum, respectively. Finally, all uncertainty relations are shown more attractive while they are expressed in terms of the symmetrized Rényi entropy.