Absztrakt: | Suppose $F$ is a function of bounded variation on $[0,2\pi]$.
A classical theorem of Fej\'er says $$\lim_{n\to\infty}{1\over
n} s_n(dF,x)=\cases {1\over \pi}\Big(F(x+)-F(x-)\Big) & {\text{
if }}0<x<2 \cr \pi, {1\over \pi}\Big(F(0+)-F(0)+F(2\pi)-F(2\pi-)\Big)
& {\text{ if }} x=0 {\text{ or }} x=2\pi \endcases\tag 1$$ where
$s_n(dF,x) $ denotes the $n-$th partial sum of the Fourier-Stieltjes
series of $F$ and $F(x+)=\lim_{\varepsilon\to 0+}F(x+\varepsilon),
F(x-)=\lim_{\varepsilon\to 0+}F(x-\varepsilon)$. The author gives
a proof of (1) for the case $x=0$ or $x=2\pi$ which is not considered
in {\it A. Zygmund}'s book [``Trigonometric series'', Vol. 1,
2nd ed. (1959; Zbl 0085.05601)]. Also, the author gives a representation
of (1) in terminology of measure. To do so, he extends the definition
of $F$ by the condition $F(x+2\pi)-F(x)=F(2\pi)-F(0)$ for all
$x\in {\Bbb R}$. Then an interval function is defined by $$\mu(F,
I)=F(b)-F(a) \quad \text{ for the finite open interval } I=(a,b).$$
Denote by $\mu_F$ the Borel measure induced by the interval function
$\mu(F,\cdot)$. Then (1) is rewritten as $$\lim_{n\to\infty}{1\over
n} s_n(dF,x)= {1\over \pi}\mu_F(\{x\})\tag 2$$ where $\{x\}$
denotes the set consisting of the single point $x$. The main
purpose of this paper is to extend (2) to the double Fourier-Stieltjes
series. The result is that for a function $F$ of bounded variation
on $[0,2\pi]\times [0,2\pi]$ in the sense of Hardy and Krause,
$$\lim_{m,n\to\infty}{1\over mn} s_{mn}(dF,x)= {1\over {\pi}^2}\mu_F(\{(x,y)\})\tag
3$$ where $s_{mn}(dF,x,y)$ denotes the $(m,n)-$th rectangular
partial sum of the Fourier-Stieltjes series of $F$ and $\mu_F(\{(x,y)\})$
denotes the value of the corresponding measure $\mu_F$ at the
single point set $\{(x,y)\}$.}
\RV{Wang Kunyang (Beijing) |