On best $𝐩$-approximation from affine subspaces: asymptotic expansion
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1077-1082.

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 $\mathrm{h}\in {ℝ}^{n}$ from a proper affine subspace K of ${ℝ}^{n}$, hK, then ${\mathrm{lim}}_{\mathrm{p}\to \infty }{h}_{p}={\mathrm{h}}_{\infty }^{*}$, 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 $\mathrm{r}\in ℕ$ there are ${\alpha }_{j}\in {ℝ}^{n}$, 1⩽jr, such that

 ${h}_{p}={\mathrm{h}}_{\infty }^{*}+\frac{{\alpha }_{1}}{\mathrm{p}-1}+\frac{{\alpha }_{2}}{{\left(\mathrm{p}-1\right)}^{2}}+\cdots +\frac{{\alpha }_{r}}{{\left(\mathrm{p}-1\right)}^{r}}+{\gamma }_{p}^{\left(\mathrm{r}\right)},$
with ${\gamma }_{p}^{\left(\mathrm{r}\right)}\in {ℝ}^{n}$ and $\parallel {\gamma }_{p}^{\left(\mathrm{r}\right)}\parallel =𝒪\left({\mathrm{p}}^{-\mathrm{r}-1}\right)$.

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 $\mathrm{h}\in {ℝ}^{n}$ par éléments d'un sous-espace affine K de ${ℝ}^{n}$, hK, alors ${\mathrm{lim}}_{\mathrm{p}\to \infty }{h}_{p}={\mathrm{h}}_{\infty }^{*}$, où ${h}_{\infty }^{*}$ est une meilleure approximation uniforme de h par éléments de K, appelée approximation uniforme stricte. Nous prouvons que hp admet un développement asymptotique du type

 ${h}_{p}={\mathrm{h}}_{\infty }^{*}+\frac{{\alpha }_{1}}{\mathrm{p}-1}+\frac{{\alpha }_{2}}{{\left(\mathrm{p}-1\right)}^{2}}+\cdots +\frac{{\alpha }_{r}}{{\left(\mathrm{p}-1\right)}^{r}}+{\gamma }_{p}^{\left(\mathrm{r}\right)},$
avec ${\alpha }_{l}\in {ℝ}^{n}$, 1⩽lr, ${\gamma }_{p}^{\left(\mathrm{r}\right)}\in {ℝ}^{n}$ et $\parallel {\gamma }_{p}^{\left(\mathrm{r}\right)}\parallel =𝒪\left({\mathrm{p}}^{-\mathrm{r}-1}\right)$.

Accepted:
Published online:
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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