Abstract
We 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)}) always possesses a zero fin S cap L^p when p<infty , but may be zero-free for fin S cap L^infty . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.
Highlights
The aim of this paper is to study the zero set of the Wigner distribution of two functions f, g ∈ L2(R), W ( f, g)(z) = Rd f (x + t 2 )g (x −)e−2π i ξ,t dt, z = (x, ξ ) ∈ R2d . (1)More precisely, we are investigating whether this zero set can be empty
The zero set of the short-time Fourier transform is important in the study of the generalized Berezin quantization and the injectivity of a general Berezin transform
The aim of this paper is to show that our belief was false by providing several examples of Wigner distributions without zeros
Summary
The aim of this paper is to study the zero set of the Wigner distribution of two functions f , g ∈ L2(R),. Results of Balk [4] yield some hints about the zero set in this case This direction does not seem to lead to new examples of zero-free Wigner distributions. We will construct a delicate example of a step function f in L∞ for which the corresponding Wigner distribution W ( f , 1(0,1)) does not have any zeros This example shows that the non-existence of zeros of the Wigner distribution may be quite subtle and may depend sensitively on integrability or smoothness properties of the function classes considered. In this part we use convexity and almost periodicity as tools.
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