On best [formula omitted]-approximation from affine subspaces: asymptotic expansion

Abstract

In this paper we consider the problem of best approximation in ℓ p ( n), 1< p⩽∞. If h p , 1< p<∞, denotes the best p-approximation of the element h∈ R n from a proper affine subspace K of R n , h∉ K, then lim p→∞h p=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 α j∈ R n , 1⩽ j⩽ r, such that h p=h ∞ ∗+ α 1 p−1 + α 2 (p−1) 2 +⋯+ α r (p−1) r +γ p (r), with γ p (r)∈ R n and ‖γ p (r)‖= O(p −r−1) . To cite this article: J.M. Quesada et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1077–1082.