Absztrakt: | Let $\{\phi\sb{i,k}\}$ be an orthonormal system on a measure
space (X,${\cal F},\mu)$. For a double orthogonal series (1)
$\sum\sp{\infty}\sb{i=0}\sum\sp{\infty}\sb{k=0}a\sb{ik}\phi\sb{ik}(x)$
the (C,$\alpha$,$\beta)$ means are defined by $$ \sigma\sp{\alpha,\beta}\sb{m,n}(x)=\frac{1}{A\sb
m\sp{\alpha}}\frac{1}{A\sb n\sp{\beta}}\sum\sp{m}\sb{i=0}\sum\sp{n}\sb{k=0}A\sp{\alpha}\sb{m-
i}A\sp{\beta}\sb{n-k}a\sb{i\quad k}\phi\sb{ik}(x) $$ where $A\sb
m\sp{\alpha}=\left( \matrix m+\alpha \\ m\endmatrix \right)$.
The main results of this paper are the following Theorem 1. If
$\alpha >1/2$, $\beta >1/2$ and (2) $\sum\sp{\infty}\sb{p=0}\sum\sp{\infty}\sb{q=0}\{\sum\sp{2\sp
p- 1}\sb{i=2\sp{p-1}}\sum\sp{2\sp q-\quad 1}\sb{k=2\sp{q-1}}a\sp
2\sb{ik}\}\sp{1/2}<\infty$ with agreement $2\sp{-1}=0$ then series
(1) is absolute $(C<\alpha,\beta)$-summable a.e. on X. Theorem
2. If $\alpha >1/2$, $\beta >1/2$ and condition (2) is not satisfied
then the double Rademacher series with coefficients $a\sb{ik}$
is not absolute (C,$\alpha$,$\beta)$-summable a.e. The author
points that these two theorems are the extensions of corresponding
one-dimensional results of other authors, and that Theorem 1
was proved by Ponomarenko and Timan in the special case of double
trigonometric series.}
\RV{K.Wang |