On best 𝐩-approximation from affine subspaces: asymptotic expansion
[Comportement asymptotique des meilleures p-approximations sur un sous-espace affine]
Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1077-1082.

Dans cette Note on considére le probléme de meilleure approximation dans ℓp(n), 1<p⩽∞. Si hp, 1<p<∞, désigne la meilleure p-approximation de h n par éléments d'un sous-espace affine K de n , hK, alors lim p h p =h * , où h * est une meilleure approximation uniforme de h par éléments de K, appelée approximation uniforme stricte. Nous prouvons que hp admet un développement asymptotique du type

h p =h * +α 1 p-1+α 2 (p-1) 2 ++α r (p-1) r +γ p (r) ,
avec α l n , 1⩽lr, γ p (r) n et γ p (r) =𝒪(p -r-1 ).

In this paper we consider the problem of best approximation in ℓp(n), 1<p⩽∞. If hp, 1<p<∞, denotes the best p-approximation of the element h n from a proper affine subspace K of n , hK, 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 there are α j n , 1⩽jr, 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) n and γ p (r) =𝒪(p -r-1 ).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02403-2
Quesada, José Marı́a 1 ; Martínez-Moreno, Juan 1 ; Navas, Juan 1

1 Departamento de Matemáticas, Universidad de Jaén, Paraje las Lagunillas, Campus Universitario, 23701 Jaén, Spain
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Quesada, José Marı́a; Martínez-Moreno, Juan; Navas, Juan. On best $ \mathbf{p}$-approximation from affine subspaces: asymptotic expansion. Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1077-1082. doi : 10.1016/S1631-073X(02)02403-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02403-2/

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