The order of magnitude of the (C,\(\alpha\) \(\geq 0,\beta \geq 0)\)-means of double orthogonal series (Q1086456)

The C(\(\alpha\),\(\beta)\)-means, which are the Cesaro means for double series, of general double orthogonal series are estimated. The author extends his earlier results on C(1,1)-means to the parameter range \(\alpha,\beta >0\) getting this way the order \(o_ x(\log \log m\cdot \log \log n)\) a.e.. This is not too surprising, however with his method he is able to derive another estimate involving quadratic averages of C(0,0)- (and even C(\(\alpha\),0), \(\alpha >-1/2)\) means with respect to the first summing index m which is remarkable, namely instead of the order \(o_ x(\log m\cdot \log n)\) that one would expect he gets \(o_ x(\log \log m\cdot \log n)\) a.e. for them.