  Single hidden layer feedforward neural networks can represent multivariate
functions that are sums of ridge functions. These ridge functions are defined
via an activation function and customizable weights. The paper deals with best
non-linear approximation by such sums of ridge functions. Error bounds are
presented in terms of moduli of smoothness. The main focus, however, is to
prove that the bounds are best possible. To this end, counterexamples are
constructed with a non-linear, quantitative extension of the uniform
boundedness principle. They show sharpness with respect to Lipschitz classes
for the logistic activation function and for certain piecewise polynomial
activation functions. The paper is based on univariate results in (Goebbels,
St.: On sharpness of error bounds for univariate approximation by single hidden
layer feedforward neural networks. Results Math 75 (3), 2020, article 109,
https://rdcu.be/b5mKH).
