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 (alpha x -theta), where alpha EA c R and theta E Theta c R, are dense in the real spaces C0 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 center dot x - theta), wE W c Rd, theta E Theta c R in C(Rd) in the topology of uniform convergence on compact subsets. (c) 2023 Elsevier Inc. All rights reserved.