Móricz Ferenc honlapja

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Cím:Extension of a theorem of Fej\'er to double Fourier-Stieltjes series.
Szerző:M\'oricz, Ferenc
Forrás:J. Fourier Anal. Appl. 7, No.6, 601-614 (2001).
Nyelv:English
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) 
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