It is known that, in general, an AP-frame is an $L^2(\mathbb{R})$-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system $\{g(t-k)e^{il(t-k)},l\in \mathbb{L}={\omega }_0\mathbb{Z},k\in \mathbb{K}=t_0\mathbb{Z}\}$ to be an $L^2(\mathbb{R})$-Frame in terms of Gaussian stationary random processes. In addition, if $X={(X(t))}_{t\in \mathbb{R}}$ is a wide sense stationary random process, we study density conditions for the associated stationary sequences $\{〈X,g_{k,l}〉,l\in \mathbb{L},k\in \mathbb{K}\}$.

中文翻译：

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AP帧和平稳随机过程

众所周知，一般来说，一个 AP 帧是一个${\mathrm{大号}}^2(\mathbb{R})$-框架，反之亦然。在这里，部分由于遍历定理，我们证明了 Gabor 系统的充分必要条件$\{G(吨-ķ)e^{\mathrm{一世}l(吨-ķ)},l\in \mathbb{大号}={\omega }_0\mathbb{Z},ķ\in \mathbb{ķ}={吨}_0\mathbb{Z}\}$成为一个${\mathrm{大号}}^2(\mathbb{R})$-根据高斯平稳随机过程的框架。此外，如果$X={(X(吨))}_{吨\in \mathbb{R}}$是广义平稳随机过程，我们研究相关平稳序列的密度条件$\{〈X,G_{ķ,l}〉,l\in \mathbb{大号},ķ\in \mathbb{ķ}\}$.