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 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 ).

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 ).

Received:
Accepted:
Published online:
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
@article{CRMATH_2002__334_12_1077_0,
     author = {Quesada, Jos\'e Mar{\i}́a and Mart{\'\i}nez-Moreno, Juan and Navas, Juan},
     title = {On best $ \mathbf{p}$-approximation from affine subspaces: asymptotic expansion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1077--1082},
     publisher = {Elsevier},
     volume = {334},
     number = {12},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02403-2},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02403-2/}
}
TY  - JOUR
AU  - Quesada, José Marı́a
AU  - Martínez-Moreno, Juan
AU  - Navas, Juan
TI  - On best $ \mathbf{p}$-approximation from affine subspaces: asymptotic expansion
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 1077
EP  - 1082
VL  - 334
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02403-2/
DO  - 10.1016/S1631-073X(02)02403-2
LA  - en
ID  - CRMATH_2002__334_12_1077_0
ER  - 
%0 Journal Article
%A Quesada, José Marı́a
%A Martínez-Moreno, Juan
%A Navas, Juan
%T On best $ \mathbf{p}$-approximation from affine subspaces: asymptotic expansion
%J Comptes Rendus. Mathématique
%D 2002
%P 1077-1082
%V 334
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02403-2/
%R 10.1016/S1631-073X(02)02403-2
%G en
%F CRMATH_2002__334_12_1077_0
Quesada, José Marı́a; Martínez-Moreno, Juan; Navas, Juan. 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. http://www.numdam.org/articles/10.1016/S1631-073X(02)02403-2/

[1] Descloux, J. Approximations in Lp and Chebychev approximations, J. Indian Soc. Appl. Math., Volume 11 (1963), pp. 1017-1026

[2] Egger, A.; Huotari, R. Rate of convergence of the discrete Pólya algorithm, J. Approx. Theory, Volume 60 (1990), pp. 24-30

[3] Fletcher, R.; Grant, J.A.; Hebden, M.D. Linear minimax approximation as the limit of best Lp-approximation, SIAM J. Numer. Anal., Volume 11 (1974) no. 1, pp. 123-136

[4] Marano, M. Strict approximation on closed convex sets, Approx. Theory Appl., Volume 6 (1990), pp. 99-109

[5] Marano, M.; Navas, J. The linear discrete Pólya algorithm, Appl. Math. Lett., Volume 8 (1995) no. 6, pp. 25-28

[6] Quesada, J.M.; Navas, J. Rate of convergence of the linear discrete Pólya algorithm, J. Approx. Theory, Volume 110 (2001), pp. 109-119

[7] J.M. Quesada, J. Martı́nez-Moreno, J. Navas, Asymptotic behaviour of best p-approximations from affine subspaces, J. Approx. Theory, submitted

[8] Rice, J.R. Tchebycheff approximation in a compact metric space, Bull. Amer. Math. Soc., Volume 68 (1962), pp. 405-410

[9] Singer, I. Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, Berlin, 1970

Cited by Sources: