Geodesic
Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity
Izvestiya. Mathematics, Tome 22 (1984) no. 2, pp. 227-245 
V. K. Dzyadyk; I. A. Shevchuk. Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity. Izvestiya. Mathematics, Tome 22 (1984) no. 2, pp. 227-245. http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a2/
@article{IM2_1984_22_2_a2,
     author = {V. K. Dzyadyk and I. A. Shevchuk},
     title = {Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity},
     journal = {Izvestiya. Mathematics},
     pages = {227--245},
     year = {1984},
     volume = {22},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a2/}
}
TY  - JOUR
AU  - V. K. Dzyadyk
AU  - I. A. Shevchuk
TI  - Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity
JO  - Izvestiya. Mathematics
PY  - 1984
SP  - 227
EP  - 245
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a2/
LA  - en
ID  - IM2_1984_22_2_a2
ER  - 
%0 Journal Article
%A V. K. Dzyadyk
%A I. A. Shevchuk
%T Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity
%J Izvestiya. Mathematics
%D 1984
%P 227-245
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a2/
%G en
%F IM2_1984_22_2_a2

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

Let $\varphi(t)$ be an arbitrary function of the type of a second modulus of continuity. It is proved that if $E\subset\mathbf R^1$, $f(x)\colon E\to\mathbf R^1$ is a given function, and \begin{equation} \biggl|f(x_2)-\frac{x_2-x_3}{x_1-x_3}f(x_1)-\frac{x_2-x_1}{x_3-x_1}f(x_3)\biggr| \leqslant2|x_1-x_2|\int_{|x_1-x_2|}^{2|x_1-x_3|}s^{-2}\varphi(s)\,ds \end{equation} for any triple of points $x_1\in E$, $x_3\in E$ and $x_2\in E\cap(x_1,x_3)$, then this function is the trace of some continuous function $\overline f\colon\mathbf R^1\to\mathbf R^1$ for which $\omega_2(\overline f,t)\leqslant A\varphi(t)$, where $A$ is an absolute constant. The function $\overline f$ is constructed by a formula which uses only the values of $\overline f$ on $E$ and the values of $\varphi(t)$. The converse of this assertion, namely, that condition (1) holds for each continuous function $f\colon\mathbf R^1\to\mathbf R^1$ on any set $E\subset \mathbf R^1$, can be verified without difficulty. Bibliography: 7 titles.

[1] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[2] Dzyadyk V. K., “K teorii priblizheniya funktsii na zamknutykh mnozhestvakh kompleksnoi ploskosti”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 134, 1975, 63–114 | Zbl

[3] Whitney H., “On functions with bounded $n$-th differences”, J. Math. Pures Appl., 36 (1957), 67–95 | MR | Zbl

[4] Marchaud A., “Sur les dérivées et sur les différences des functions de variables réelles”, J. Mathem. Pures et Appl., 6 (1927), 337–425 | Zbl

[5] Dzyadyk V. K., “O priblizhenii funktsii obyknovennymi mnogochlenami na konechnom otrezke veschestvennoi osi”, Izv. AN SSSR. Ser. matem., 22:3 (1958), 337–354 | MR

[6] Bari N. K., Stechkin S. B., “Nailuchshee priblizhenie i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. matem. o-va, 5, 1956, 483–552 | MR

[7] Shevchuk I. A., “Nekotorye zamechaniya o funktsiyakh tipa modulya nepreryvnosti poryadka $k\geqslant2$”, Voprosy teorii priblizheniya funktsii i ee prilozhenii, In-t matem. AN USSR, Kiev, 1976, 194–199