Geodesic
A multidimensional analog of a theorem of Whitney
Sbornik. Mathematics, Tome 11 (1970) no. 2, pp. 157-170 
Yu. A. Brudnyi. A multidimensional analog of a theorem of Whitney. Sbornik. Mathematics, Tome 11 (1970) no. 2, pp. 157-170. http://geodesic.mathdoc.fr/item/SM_1970_11_2_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The following theorem is proved: Theorem. {\it Let $f\in L_p(\Omega)$, where $\Omega$ is a convex domain in $R^n$. Then $$ \inf_l\|f-l\| _{L_p(\Omega)}\leqslant w\sup_h\|\Delta_h^kf\|, $$ where the $\inf$ on the left is taken over all degree $k-1$ polynomials, and the $L_p$ norm on the right is taken over the set in which the $k$th difference $\Delta_h^kf$ is defined. The constant $w$ depends only on $k,n$, and the ratio of the diameter of $\Omega$ to its width}. H. Whitney proved this theorem in the case $p=\infty$ and $\Omega=[0,1]$. As a corollary, it is proved that the $k$-modulus of continuity dominates any “deviation”, constructed with the help of a measure with compact support, orthogonal to polynomials of degree $k-1$. Bibliography: 10 titles.

[1] H. Whitney, “On functions with bounded $n$-th differences”, J. Math. Pures AppL., 9:36 (1967), 67–95 | MR

[2] A. P. Kalderon, “Promezhutochnye prostranstva i interpolyatsiya; kompleksnyi metod”, Matematika, 9:3 (1965), 56–129

[3] Yu. A. Brudnyi, Issledovaniya po teorii lokalnogo nailuchshego priblizheniya, Dissertatsiya, Dnepropetrovsk, 1965

[4] Yu. A. Brudnyi, “Issledovanie svoistv periodicheskikh funktsii mnogikh peremennykh”, Uspekhi matem. nauk, XX:5(95) (1965), 276–279

[5] Yu. A. Brudnyi, “Raznostnye kriterii suschestvovaniya proizvodnykh v $L_p$”, Matem. sb., 73(115) (1967), 42–64 | MR | Zbl

[6] A. F. Timan, Teoriya priblizheniya funktsii, Fizmatgiz, Moskva, 1960

[7] M. F. Timan, “Nekotorye tozhdestva dlya funktsii mnogikh peremennykh”, DAN SSSR, 178:1 (1968), 40–42 | MR | Zbl

[8] N. Danford, Dzh. Shvarts, Teoriya operatorov, t. I, IL, Moskva, 1962

[9] A. N. Kolmogorov, V. M. Tikhomirov, “$\varepsilon$-entropiya i $\varepsilon$-emkost mnozhestv v funktsionalnykh prostranstvakh”, Uspekhi matem. nauk, XIX:2(86) (1959), 3–86 | MR

[10] H. S. Shapiro, “A tauberian theorem related to approximation theory”, Acta Matrix, 120 (1968), 279–292 | DOI | MR | Zbl