Heisenberg-type inequalities for the Weinstein operator

Abstract

The aim of this paper is to prove new uncertainty principles for the Weinstein operator. The first of these results is a sharp Heisenberg-type inequality for the Weinstein transform, that is, for s≥1 and f∈Lα2(R+d), ‖|x|sf‖Lα2(R+d)‖|ξ|sFW(f)‖Lα2(R+d)≥α+(d+1)2s∥f∥Lα2(R+d)2. The second result states that the previous inequality can be refined for an orthonormal basis for Lα2(R+d), that is, if {φn}n=1∞ is an orthonormal basis for Lα2(R+d), then supn(∥|x|sφn∥Lα2(R+d)∥|ξ|sFW(φn)∥Lα2(R+d))=∞. As a side result, we prove a new version of Heisenberg's uncertainty inequality for the Weinstein–Gabor transform, which states that the Weinstein–Gabor transform of a nonzero function with respect to a nonzero window function cannot be time and frequency concentrated around zero.