Geodesic
Best approximation to monomials on a cube
Sbornik. Mathematics, Tome 199 (2008) no. 8, pp. 1251-1262 
V. A. Yudin. Best approximation to monomials on a cube. Sbornik. Mathematics, Tome 199 (2008) no. 8, pp. 1251-1262. http://geodesic.mathdoc.fr/item/SM_2008_199_8_a5/
@article{SM_2008_199_8_a5,
     author = {V. A. Yudin},
     title = {Best approximation to monomials on a~cube},
     journal = {Sbornik. Mathematics},
     pages = {1251--1262},
     year = {2008},
     volume = {199},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_8_a5/}
}
TY  - JOUR
AU  - V. A. Yudin
TI  - Best approximation to monomials on a cube
JO  - Sbornik. Mathematics
PY  - 2008
SP  - 1251
EP  - 1262
VL  - 199
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_2008_199_8_a5/
LA  - en
ID  - SM_2008_199_8_a5
ER  - 
%0 Journal Article
%A V. A. Yudin
%T Best approximation to monomials on a cube
%J Sbornik. Mathematics
%D 2008
%P 1251-1262
%V 199
%N 8
%U http://geodesic.mathdoc.fr/item/SM_2008_199_8_a5/
%G en
%F SM_2008_199_8_a5

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

The paper considers a multivariate analogue of the Chebyshev problem on the cube concerning the construction of polynomials of least deviation from zero. A classification of monomials possessing a unique polynomial of best approximation in the space of continuous functions on the unit cube in $\mathbb R^n$ is given. Precise solutions in some weighted spaces $L_p$ are found.

[1] P. L. Chebyshev, Sobranie sochinenii, Izd-vo AN SSSR, M.–L., 1944–1951 | MR | Zbl

[2] E. I. Zolotarev, “Sur un certain minimum”, Polnoe sobranie sochinenii, t. 1, Izd-vo AN SSSR, L., 1931, 138–153 | Zbl

[3] S. N. Bernshtein, Polnoe sobranie sochinenii, t. 1, 2, Izd-vo AN SSSR, M., 1952–1954 | MR | MR | Zbl

[4] N. I. Akhiezer, Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR | Zbl

[5] J. M. Sloss, “Chebyshev approximation to zero”, Pacific J. Math., 15:1 (1965), 305–313 | MR | Zbl

[6] H. Ehlich, K. Zeller, “Čebyšev-Polynome in mehreren Veränderlichen”, Math Z., 93:2 (1966), 142–143 | DOI | MR | Zbl

[7] J. Fromm, “$L_1$-approximation to zero”, Math. Z., 151:1 (1976), 31–33 | DOI | MR | Zbl

[8] M. Reimer, “On multivariate polynomials of least deviation from zero on the unit cube”, J. Approx. Theory, 23:1 (1978), 65–69 | DOI | MR | Zbl

[9] W. B. Gearhart, “Some Chebyshev approximations by polynomials in two variables”, J. Approximation Theory, 8:3 (1973), 195–209 | DOI | MR | Zbl

[10] V. A. Yudin, “On polynomials of best approximation”, Math. Notes, 82:3–4 (2007), 564–568 | DOI | MR

[11] T. J. Rivlin, H. S. Shapiro, “A unified approach to certain problems of approximation and minimization”, J. Soc. Indust. Appl. Math., 9:4 (1961), 670–699 | DOI | MR | Zbl