Geodesic
On the relation between the Jackson and Jung constants of the spaces $L_ p$
Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 828-836 
V. I. Ivanov. On the relation between the Jackson and Jung constants of the spaces $L_ p$. Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 828-836. http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a2/
@article{MZM_1995_58_6_a2,
     author = {V. I. Ivanov},
     title = {On~the relation between the {Jackson} and {Jung} constants of the spaces $L_ p$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {828--836},
     year = {1995},
     volume = {58},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a2/}
}
TY  - JOUR
AU  - V. I. Ivanov
TI  - On the relation between the Jackson and Jung constants of the spaces $L_ p$
JO  - Matematičeskie zametki
PY  - 1995
SP  - 828
EP  - 836
VL  - 58
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a2/
LA  - ru
ID  - MZM_1995_58_6_a2
ER  - 
%0 Journal Article
%A V. I. Ivanov
%T On the relation between the Jackson and Jung constants of the spaces $L_ p$
%J Matematičeskie zametki
%D 1995
%P 828-836
%V 58
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a2/
%G ru
%F MZM_1995_58_6_a2

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

For any infinitely metrizable compact Abelian group $G$, $1\leqslant p\leqslant q<\infty$, $n\in\mathbb N$, the following relations are proved: $$ K_{pq}(G,n,G)=d_{pq}(G,n,G)=J(L_p(G),L_q(G))=\varkappa_{pq}, $$ where $K_{pq}(G,n,G)$ is the largest Jackson constant in the approximation of the system of characters by polynomials of order $n$, $d_{pq}(G,n,G)$ is the best Jackson constant, $J(L_p(G),L_q(G))$ is the Jung constant of the pair of real spaces $(L_p(G),L_q(G))$, and $$ \begin{aligned} \varkappa_{pq}^q&=\sup\biggl\{\inf_c\int_0^1|f(x)-c|^q\,dx \\ &\qquad\qquad\times\biggl|\int_0^1\int_0^1|f(x)-f(y)|\biggr|^p\,dx\,dy\le1,\ f\in L_q[-1,1]\biggr\}. \end{aligned} $$

[1] Ivanov V. I., “Ob otsenke snizu konstanty v neravenstve Dzheksona v raznykh $L_p$-normakh”, Matem. zametki, 52:3 (1992), 48–62 | MR | Zbl

[2] Khyuitt E., Ross K. A., Abstraktnyi garmonicheskii analiz, T. 1, Mir, M., 1975

[3] Amir D., “On Jung's constants and related constants in normed linear spaces”, Pacific J. Math., 118:1 (1985), 1–15 | MR | Zbl

[4] Ivanov V. I., Pichugov S. A., “Konstanta Yunga $l_p^n$-prostranstv”, Matem. zametki, 48:4 (1990), 37–47 | MR | Zbl

[5] Pichugov S. A., “Konstanta Yunga prostranstva $L_p$”, Matem. zametki, 43:5 (1988), 604–614 | MR

[6] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | Zbl

[7] Fan K., “Fixed point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad. Sci. USA, 38 (1952), 121–126 | DOI | MR | Zbl

[8] Chernykh N. I., “Neravenstvo Dzheksona v $L_p(0,2\pi )$ s tochnoi konstantoi”, Tr. MIAN SSSR, 198, Nauka, M., 1992, 232–241 | Zbl

[9] Ivanov V. I., “O module nepreryvnosti v $L_p$”, Matem. zametki, 41:5 (1987), 682–686 | MR | Zbl

[10] Routledge N., “A result in Hilbert space”, Qart. J. Math., 3 (1952), 12–18 | DOI | MR | Zbl

[11] Berdyshev V. I., “Svyaz mezhdu neravenstvom Dzheksona i odnoi geometricheskoi zadachei”, Matem. zametki, 3:3 (1968), 327–338 | MR | Zbl

[12] Korneichuk N. P., Splainy v teorii priblizheniya, Nauka, M., 1984

[13] Kholl M., Kombinatorika, Mir, M., 1970