Extension of functions that are traces on an arbitrary subset of the line of functions with given second modulus of continuity

V. K. Dzyadyk ; I. A. Shevchuk

Izvestiya. Mathematics , Tome 22 (1984) no. 2, pp. 227-245

Résumé

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.