Let
sO( Î - s log(1/ Î ))\mathcal{O}( \in ^{ - s} log(1/ \in ))
if a spline-like localization is required. This cannot be improved even if one allows different neurons to evaluate different activation functions, even depending upon the target function. Nevertheless, for any
O( Î - s - d )\mathcal{O}( \in ^{ - s - \delta } )
neurons can be constructed to provide this order of approximation, with localization. Analogous results are also valid for otherL
p
norms.Keywords Neural networks - Sobolev spaces - spline approximation - ridge functions
AMS subject classification 41A63 - 41A30 - 94C99
The research of this author was supported by NSF Grant # DMS 92-0698.
The research of this author was supported, in part, by AFOSR Grant #F49620-93-1-0150 and by NSF Grant #DMS 9404513.