Representation of~measurable functions almost everywhere by convergent series

F. G. Arutyunyan

Sbornik. Mathematics, Tome 19 (1973) no. 4, pp. 469-508

Résumé

In this paper it is proved that for a certain class of systems $\{\varphi _k\}$ (systems of type $(\mathrm{X})$) one may construct a series \begin{equation} \sum^\infty_{k=1}a_k\varphi_k(t),\qquad t\in[0,1], \end{equation} having the following properties: 1) $\lim_{k\to\infty}a_k\varphi_k(t)=0$ uniformly on the interval $[0,1]$. 2) For any measurable function $f(t)$ on the interval $[0,1]$ and for any number $N$, one can find a partial series $$ \sum^\infty_{k=1}a_{n_k}\varphi_{n_k}(t),\qquad(N\cdots), $$ from (1) which converges to $f(t)$ almost everywhere on the set where $f(t)$ is finite, and converges to $f(t)$ in measure on $[0,1]$. 3) If, in addition, the functions $\varphi_k$ ($k\geqslant1$) and $f$ are piecewise continuous and $\inf_{t\in[0,1]}f(t)>0$, then $$ \sum^\infty_{k=1}a_{n_k}\varphi_{n_k}(t)\qquad\text{for all $t\in[0,1]$ and $m\geqslant1$}. $$ It is shown that systems of type $(\mathrm X)$ include, for example, trigonometric systems, the systems of Haar and Walsh, indexed in their original or a different order, any basis of the space $C(0,1)$, and others. Bibliography: 19 titles.