Best approximation by ridge functions in $L_p$-spaces - Maiorov, V. E. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
We study the approximation of the classes of functions by the manifold $R_n$ formed by all possible linear combinations of $n$ ridge functions of the form $r(a · x))$. It is proved that, for any $1 ≤ q ≤ p ≤ ∞$, the deviation of the Sobolev class $W^r_p$ from the set $R_n$ of ridge functions in the space $L_q (B^d)$ satisfies the sharp order $n^{-r/(d-1)}$.