The order of magnitude of the arithmetic means of double orthogonal series (Q1087768)

With the partial sums \(\{s_{ik}(x)\}\) of the double orthogonal series \(\sum \sum a_{ik}\phi_{ik}(x)\) the author considers the first arithmetic means \(\sigma^{11}_{mn}(x)=\frac{1}{m\cdot n}\sum^{m}_{i=1}\sum^{n}_{k=1}s_{ik}(x).\) His main result shows that under condition \(\sum \sum a^ 2_{ik}<\infty\) then \(\sigma^{11}_{mn}(x)=o_ x(\log \log (m+3)\log \log (n+3))\) a.e. Besides that similar results are deduced for means \(\sigma^{10}_{mn}(x)\) (resp. \(\sigma^{01}_{mn}(x))\) formed only with respect to m (resp. n.) The estimates must not be weakened essentially. In general these results stress that double orthogonal series behave like products of single orthogonal series even in the field of summability.