Zeros of the Wigner distribution and the short-time Fourier transform

Abstract

AbstractWe study the question under which conditions the zero set of a (cross-) Wigner distribution W(f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson’s theorem for the positivity of the Wigner distribution and to Hardy’s uncertainty principle. We then construct a class of step functions S so that the Wigner distribution $$W(f,\mathbf {1}_{(0,1)})$$ W ( f , 1 ( 0 , 1 ) ) always possesses a zero $$f\in S \cap L^p$$ f ∈ S ∩ L p when $$p<\infty $$ p < ∞ , but may be zero-free for $$f\in S \cap L^\infty $$ f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.