Uncertainty principles on certain Lie groups
The authors show how various uncertainty principles can be formulated in the case of some locally compact groups including \(\mathbb{R}^n\), the Heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane. A theorem of Hardy, which is one way of formulating the uncertainty principle for the Fourier transform \(\widehat f\), is the following Theorem (Hardy): Suppose \(f\) is a measurable function on \(\mathbb{R}\) such that (1) \(|f(x)|\leq Ce^{-\alpha x^2}\), and \(|\widehat f(\xi) |\leq Ce^{-\beta \xi^2}\), \(x,\xi\in \mathbb{R}\), \(\alpha, \beta>0\). If \(\alpha\beta > {1\over 4}\) then \(f=0\) a.e. If \(\alpha \beta< {1\over 4}\) then there are infinitely many linearly independent functions satisfying (1) and if \(\alpha\beta = {1\over 4}\) then \(f(x)=Ce^{-\alpha x^2}\). -- The authors prove an analogue of Hardy's theorem for \(\mathbb{R}^n\) \((n\geq 2)\), the Heisenberg group and the Euclidean motion group of the plane, and show that though the exact analogue for the reduced Heisenberg group fails, a slightly modified version continues to hold. Furthermore, they show another way of formulating the uncertainty principle for the Fourier transform on the Heisenberg group in terms of the supports of the function \(f\) and its Fourier transform \(\widehat f\). They formulate and prove a local uncertainty inequality. From this uncertainty inequality they deduce a global inequality similar to the classical Heisenberg-Pauli-Weyl uncertainty inequality.
Heisenberg-Pauli-Weyl uncertainty inequality
scientific article; zbMATH DE number 903346
scientific article