Geodesic
A~nowhere dense space of linear superpositions of functions of several variables
Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 807-837.

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

Let $\mathbf I^3$ be the unit cube of three-dimensional space $R^3$, and let $\Phi_i(x)$, $i=1,\dots,n$, be mappings $\Phi_i\colon\mathbf I^3\to R^2$ of class $C_2$. We prove that the set of functions $F(x)$ on $\mathbf I^3$ which can be represented in the form $$ F(x)=\sum_{i=1}^n(\chi_i\circ\Phi_i)(x), $$ where the $\chi_i(u_1,u_2)$ are arbitrary continuous functions, $\chi_i\colon R^2\to R$, is nowhere dense in $\mathscr L_2(\mathbf I^3)$. 
@article{IM2_1972_6_4_a7,
     author = {B. L. Fridman},
     title = {A~nowhere dense space of linear superpositions of functions of several variables},
     journal = {Izvestiya. Mathematics },
     pages = {807--837},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/}
}
TY  - JOUR
AU  - B. L. Fridman
TI  - A~nowhere dense space of linear superpositions of functions of several variables
JO  - Izvestiya. Mathematics 
PY  - 1972
SP  - 807
EP  - 837
VL  - 6
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/
LA  - en
ID  - IM2_1972_6_4_a7
ER  - 
%0 Journal Article
%A B. L. Fridman
%T A~nowhere dense space of linear superpositions of functions of several variables
%J Izvestiya. Mathematics 
%D 1972
%P 807-837
%V 6
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/
%G en
%F IM2_1972_6_4_a7
B. L. Fridman. A~nowhere dense space of linear superpositions of functions of several variables. Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 807-837. http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/

[1] Kolmogorov A. N., “O predstavlenii nepreryvnykh funktsii neskolkikh peremennykh v vide superpozitsii nepreryvnykh funktsii odnogo peremennogo i slozheniya”, Dokl. AN SSSR, 114:5 (1957), 953–956 | MR

[2] Fridman B. L., “Uluchshenie gladkosti funktsii v teoreme A. N. Kolmogorova o superpozitsiyakh”, Dokl. AN SSSR, 177:5 (1967), 1019–1022 | MR | Zbl

[3] Vitushkin A. G., Khenkin G. M., “Lineinye superpozitsii funktsii”, Uspekhi matem. nauk, XXII:1(133) (1967), 77–124 | MR

[4] Vitushkin A. G., “O vozmozhnosti predstavleniya funktsii superpozitsiyami funktsii ot menshego chisla peremennykh”, Tr. Mezhdunarodnogo kongressa matematikov (M., 1966), Mir, M., 1968, 322–328

[5] Fridman B. L., “Otsenka razmernosti prostranstva nulei lineinykh superpozitsii”, Matem. sb., 82(124):1(5) (1970), 111–125 | MR | Zbl