We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of $L_2([0,1])$ based on the Gershgorin theorem. We also generalize this system to higher dimensions $d>1$ by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of $L_2([0,1]^d)$ can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of $d$, making it an attractive building block regarding future multivariate analysis of neural networks.

翻译：我们考虑了Daubechies、DeVore、Foucart、Hanin和Petrova近期提出的分段线性函数类三角系统。基于Gershgorin定理，我们给出了该系统构成$L_2([0,1])$空间Riesz基的替代性证明。我们还通过一种避免使用（张量）积的构造方法，将该系统推广至高维情形$d>1$。由此，新$L_2([0,1]^d)$空间Riesz基中的函数可便捷地通过神经网络表示。此外，该系统的Riesz常数与维度$d$无关，这使其成为未来神经网络多变量分析中具有吸引力的基本构建模块。