On best $𝐩$-approximation from affine subspaces: asymptotic expansion

Quesada, José Marı́a; Martínez-Moreno, Juan; Navas, Juan

Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1077-1082.

Dans cette Note on considére le probléme de meilleure approximation dans ℓ_p(n), 1<p⩽∞. Si h_p, 1<p<∞, désigne la meilleure p-approximation de $\mathrm{h}\in {ℝ}^n$ par éléments d'un sous-espace affine K de ${ℝ}^n$, h∉K, alors ${\lim }_{\mathrm{p}\to \infty }{h}_p={\mathrm{h}}_{\infty }^{*}$, où $h_{\infty }^{*}$ est une meilleure approximation uniforme de h par éléments de K, appelée approximation uniforme stricte. Nous prouvons que h_p admet un développement asymptotique du type

$$h_p={\mathrm{h}}_{\infty }^{*}+\frac{{\alpha }_1}{\mathrm{p}-1}+\frac{{\alpha }_2}{{(\mathrm{p}-1)}^2}+\cdots +\frac{{\alpha }_r}{{(\mathrm{p}-1)}^r}+{\gamma }_p^{\left(\mathrm{r}\right)},$$

avec ${\alpha }_l\in {ℝ}^n$, 1⩽l⩽r, ${\gamma }_p^{\left(\mathrm{r}\right)}\in {ℝ}^n$ et $\parallel {\gamma }_p^{\left(\mathrm{r}\right)}\parallel =𝒪\left({\mathrm{p}}^{-\mathrm{r}-1}\right)$.

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 $\mathrm{h}\in {ℝ}^n$ from a proper affine subspace K of ${ℝ}^n$, h∉K, then ${\lim }_{\mathrm{p}\to \infty }{h}_p={\mathrm{h}}_{\infty }^{*}$, where $h_{\infty }^{*}$ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all $\mathrm{r}\in \mathbb{N}$ there are ${\alpha }_j\in {ℝ}^n$, 1⩽j⩽r, such that

$$h_p={\mathrm{h}}_{\infty }^{*}+\frac{{\alpha }_1}{\mathrm{p}-1}+\frac{{\alpha }_2}{{(\mathrm{p}-1)}^2}+\cdots +\frac{{\alpha }_r}{{(\mathrm{p}-1)}^r}+{\gamma }_p^{\left(\mathrm{r}\right)},$$

with ${\gamma }_p^{\left(\mathrm{r}\right)}\in {ℝ}^n$ and $\parallel {\gamma }_p^{\left(\mathrm{r}\right)}\parallel =𝒪\left({\mathrm{p}}^{-\mathrm{r}-1}\right)$.

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