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

We find sufficient conditions on a function f to ensure that sums of functions of the form f(αx−θ), where α∈A⊂R and θ∈Θ⊂R, are dense in the real spaces C0 and Lp 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⋅x−θ), w∈W⊂Rd, θ∈Θ⊂R in C(Rd) in the topology of uniform convergence on compact subsets.