Asymptotic Behaviour of Best lp-Approximations from Affine Subspaces

In this paper we consider the problem of best approximation in lpn, 1 < p ≤ ∞. If hp, 1 < p < ∞, denotes the best lp-approximation of the element h ∈ Rn from a proper affine subspace K of Rn, h ∉ K, then limp→∞hp = h*∞ where h*∞ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r ∈ N there are αj ∈ Rn, 1 ≤ j ≤ r, such that hp = h*∞ + α1/p-1 + α2/(p-1)2 + ... + αr/(p-1)r + γpr, with γp(r) ∈ Rn and ||γp(r)|| = O(p-r-1).