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Representation of measurable functions almost everywhere by convergent series
F. G. Arutyunyan
Abstract:
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
$$
\sum^\infty_{k=1}a_{n_k}\varphi_{n_k}(t),\qquad(N<n_1<n_2<\cdots),
$$
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.
UDC:
517.512
MSC: Primary
42A56,
42A60; Secondary
42A20 Received: 06.09.1972
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