Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ C d (R d), with lower-order partials vanishing at infinity and dth-order partials vanishing as x −(d+1+ε) , ε > 0, on any domain ⊂ R d with unit… (More)
A b s t r a c t I t i s s h o w n t h a t i n a B a n a c h s p a c e
In Lp-spaces with p an integer from [1, infinity) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p an integer from (1, infinity) such best approximation is not unique and cannot be continuous.
It is shown that for any positive integer n and any function in Lp([0, 1] d) with p ∈ [1, ∞) there exists a best approximation by linear combinations of n characteristic functions of half-spaces. Further, sequences of such linear combinations converging in distance to the best approximation distance have subsequences converging to the best approximation,… (More)
Upper bounds on variation with respect to half-spaces, in preparation.