Approximation by sums of shifts and dilations of a single function and neural networks

Abstract

We find sufficient conditions on a function $f$ to ensure that sums of functions of the form $f(\alpha x-\theta )$, where $\alpha \in A\subset R$ and $\theta \in \Theta \subset R$, are dense in the real spaces $C_0$ and $L_p$ on the real line or its compact subsets. That is, we consider linear combinations in which all coefficients are 1. As a corollary we deduce results on density of sums of functions $f(w\cdot x-\theta )$, $w\in W\subset R^d$, $\theta \in \Theta \subset R$ in $C\left(R^d\right)$ in the topology of uniform convergence on compact subsets.