TDI-subspaces of \(C(\mathbb{R}^ d)\) and some density problems from neural networks
Space \(W\subset C(\mathbb{R}^d)\) is said to be TDI if it is invariant under actions of the group generated by translations and dilations (in each coordinates) of \(\mathbb{R}^d\). Let \(M_f\) be the closure of the smallest TDI-space which contains \(f\). Theorem 1. \(M_f\neq C(\mathbb{R}^d)\Leftrightarrow\) there is \(\alpha\in\mathbb{Z}^d_+\) such that \(D^\alpha f=0\) (in the weak sense). Let \(A_\alpha:=\overline{\text{span}}\{x^\beta; \beta\leq\alpha\}\); here some coordinates of \(\alpha\) may be infinite but all \(\beta\in\mathbb{Z}^d_+\). Theorem 2. Every TDI-space is a finite sum of the \(A_\alpha\).
scientific article; zbMATH DE number 894208