Comptes Rendus
no. 12
On best 𝐩-approximation from affine subspaces: asymptotic expansion
[Comportement asymptotique des meilleures p-approximations sur un sous-espace affine]
José Marı́a Quesada1 ; Juan Martínez-Moreno1 ; Juan Navas1
1 Departamento de Matemáticas, Universidad de Jaén, Paraje las Lagunillas, Campus Universitario, 23701 Jaén, Spain
Comptes Rendus. Mathématique, Volume 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∈ℝ n par éléments d'un sous-espace affine K de ℝ n , h∉K, 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 ∈ℝ n , 1⩽l⩽r, γ p (r) ∈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∈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 ).

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

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

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