Asymptotic behaviour of best l(p)-approximations from affine subspaces

摘要

In this paper we consider the problem of best approximation in l(p)(n), 1 < p less than or equal to infinity. If h(p) 1 < p < infinity, denotes the best l(p)-approximation of the element h epsilon R-n from a proper affine subspace K of R-n, h epsilon K, then lim(p-->)infinityh(p) = h(infinity)(*) where h(infinity)(*) is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r epsilon N there are alpha(j) epsilon R-n, 1 less than or equal to j less than or equal to r, such that h(p) = h(infinity)(*) + alpha(1)/p - 1 + alpha(2)/(p - 1)(2) + (...) + alpha(r)/(p - 1)(r) + gamma(p)((r)) with gamma(p)((r)) epsilon R-n and parallel togamma(p)((r))parallel to = O(p(-r-1)). (C) 2002 Elsevici Science (USA). [References: 8]