Polar Wavelet Transform and the Associated Uncertainty Principles

Abstract

The polar wavelet transform– a generalized form of the classical wavelet transform has been extensively used in science and engineering for finding directional representations of signals in higher dimensions. The aim of this paper is to establish new uncertainty principles associated with the polar wavelet transforms in $L^{2}(\mathbb R^{2})$. Firstly, we study some basic properties of the polar wavelet transform and then derive the associated generalized version of Heisenberg–Pauli–Weyl inequality. Finally, following the idea of Beckner (Proc. Amer. Math. Soc. 123, 1897–1905 1995), we drive the logarithmic version of uncertainty principle for the polar wavelet transforms in $L^{2}(\mathbb R^{2})$.