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 \(f\in S \cap L^p\) when \(p<\infty \), but may be zero-free for \(f\in S \cap L^\infty \). The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

中文翻译：

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Wigner分布的零点和短时傅立叶变换

我们研究一个问题，在这种情况下，（交叉）Wigner分布W（f，  g）的零集或短时傅立叶变换为空。当f和g均为广义高斯时，就是这种情况，但是我们将构造由指数函数及其卷积组成的不那么明显的示例。结果需要完全正函数，贝塞尔函数和Hurwitz多项式的理论中的元素。无零维格纳分布的问题也与维格纳分布的正定性的哈德森定理和哈迪的不确定性原理有关。然后，我们构造一类阶跃函数S，以使Wigner分布\（W（F，\ mathbf {1} _ {（0,1）}）\）始终具有零\（F \ S中\帽L ^ p \）时\（P <\ infty \） ，但对于\（f \ in S \ cap L ^ \ infty \）可能为零。这些示例表明，维格纳分布的零问题可能非常微妙，并且涉及多个分析分支。