A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [4]. This answered pioneering work by Flandrin [10], who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises. Our main observation is a correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. Moreover, we discuss subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces and its finite dimensional approximations.

高斯分析函数（GAF）家族最近已与高斯白噪声的Gabor变换相关联[4]。这回答了Flandrin [10]的开创性工作，Flandrin观察到白噪声的Gabor变换的零点具有正态分布，并提出了基于频谱图零点的滤波算法。在本文中，我们系统地研究了GAF与一类高斯白噪声的时频变换之间的联系。我们的主要观察结果是对（变换，GAF）与经典正交多项式的生成函数之间的对应关系。这涵盖了一些经典的时频变换，例如Gabor变换和Daubechies-Paul小波变换。它还推出了新的开窗离散傅立叶变换，可将白噪声映射到基本GAF。此外，