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Cím:On the $\vert C,\alpha >1/2,\beta >1/2\vert$-summability of double orthogonal series.
Szerző:M\'oricz, F.
Forrás:Acta Sci. Math. 48, 325-338 (1985).
Nyelv:English
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 
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