Geodesic
Approximation error for linear polynomial interpolation on $n$-simplices
Matematičeskie zametki, Tome 60 (1996) no. 4, pp. 504-510.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $W_n^2M$ be the class of functions $f\colon\Delta_n\to\mathbb R$ (when ($\Delta_n$ is an $n$-simplex) with bounded second derivative (whose absolute value does not exceed $M>0$) along any direction at an arbitrary point of the simplex $\Delta_n$. Let $P_{1,n}(f;x)$ be the linear polynomial interpolating $f$ at the vertices of the simplex. We prove that there exists a function $g\in W_n^2M$ such that for any $f\in W_n^2M$ and any $x\in\Delta_n$ one has $$ |f(x)-P_{1,n}(f;x)|\leqslant g(x). $$ 
@article{MZM_1996_60_4_a2,
     author = {Yu. A. Kilizhekov},
     title = {Approximation error for linear polynomial interpolation on $n$-simplices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {504--510},
     publisher = {mathdoc},
     volume = {60},
     number = {4},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/}
}
TY  - JOUR
AU  - Yu. A. Kilizhekov
TI  - Approximation error for linear polynomial interpolation on $n$-simplices
JO  - Matematičeskie zametki
PY  - 1996
SP  - 504
EP  - 510
VL  - 60
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/
LA  - ru
ID  - MZM_1996_60_4_a2
ER  - 
%0 Journal Article
%A Yu. A. Kilizhekov
%T Approximation error for linear polynomial interpolation on $n$-simplices
%J Matematičeskie zametki
%D 1996
%P 504-510
%V 60
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/
%G ru
%F MZM_1996_60_4_a2
Yu. A. Kilizhekov. Approximation error for linear polynomial interpolation on $n$-simplices. Matematičeskie zametki, Tome 60 (1996) no. 4, pp. 504-510. http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/

[1] Subbotin Yu. N., “Zavisimost otsenok mnogomernoi kusochno-polinomialnoi approksimatsii ot geometricheskikh kharakteristik triangulyatsii”, Tr. MIAN, 189, Nauka, M., 1989, 117–137 | MR

[2] Reztsov A. V., Optimalnye kubaturnye formuly na klassakh differentsialnykh funktsii, Dep. VINITI, No 4227–V89

[3] Subbotin Yu. N., “Pogreshnost approksimatsii interpolyatsionnymi mnogochlenami malykh stepenei na $n$-simpleksakh”, Matem. zametki, 48:4 (1990), 88–100 | MR

[4] Borwein J., Keener L., “The Hausdorff meyric and Chebyshev centers”, J. Approxim. Theory, 28 (1955), 366–376 | DOI