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Cím:Approximation theorems for double orthogonal series.
Szerző:M\'oricz, F.
Forrás:J. Approximation Theory 42, 107-137 (1984).
Nyelv:English
Absztrakt:Let $\{\phi\sb{ik}(x): i,k=1,2,...\}$ be a double orthonormal system on a positive measure space, and $\{a\sb{ik}\}$ a double sequence of real numbers for which $\sum\sp{\infty}\sb{i=1}\sum\sp{\infty}\sb{k=1}a\sp 2\sb{ik}<\infty$. Then the sum f(x) of the double orthogonal series $\sum \sum a\sb{ik}\phi\sb{ik}(x)$ exists in the sense of $L\sp 2$-metric. If, in addition, $(*)\quad \sum \quad \sum a\sp 2\sb{ik}\kappa\sp 2(i,k)<\infty$ with an appropriate double sequence $\{ \kappa$ (i,k)$\}$ of positive numbers, then a rate of approximation to f(x) can be obtained by the rectangular partial sums $s\sb{mn}(x)=\sum\sp{m}\sb{i=1}\sum\sp{n}\sb{k=1}a\sb{ik}\phi\sb{ik}(x),$ by the first arithmetic means $\sigma\sb{mn}(x)=(1/mn)\sum\sp{m}\sb{i=1}\sum\sp{n}\sb{k=1}s\sb{ik}(x),$ etc. One of the main results reads as follows: Theorem 4. If $\{\lambda\sb j(m): m=1,2,...\}$ is a nondecreasing sequence of positive numbers such that $\lambda\sb j(2m)\le C\lambda\sb j(m)$ with a constant $C<2$ for $m\ge m\sb j (j=1,2)$ and relation (*) is satisfied with $\kappa(i,k)=\log \log(i+3)\log \log(k+3)\cdot \max \{\lambda\sb 1(i),\lambda\sb 2(k)\},$ then $\min \{\lambda\sb 1(m),\lambda\sb 2(n)\}\vert \sigma\sb{mn}(x)-f(x)\vert \to 0$ a.e. as mi$n\{$ m,$n\} \to \infty$. 
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