Rate of Cesàro summability of double orthogonal series (Q5916161)

Let \(\{\phi_{i_1,i_2}(x): i_1,i_2= 0,1,2,\dots\}\) be a double ONS on the unit interval \((0,1)\) and consider the double orthogonal series \[ \sum^\infty_{i_1= 0} \sum^\infty_{i_2= 0} a_{i_1,i_2} \phi_{i_1,i_2}(x),\tag{\(*\)} \] where \[ \sum^\infty_{i_1= 0} \sum^\infty_{i_2= 0} a^2_{i_1,i_2} \lambda^2_1(i_1) \lambda^2_2(i_2)< \infty \] and \(\lambda_j(i_j)\uparrow\infty\), \(j= 1,2\). Let \(\alpha_1,\alpha_2> 0\) and denote by \(\sigma^{\alpha_1, \alpha_2}_{n_1,n_2}(x)\) the Cesàro means of the rectangular partial sums of the orthogonal series \((*)\). The order of magnitude of the deviation \(|f(x)- \sigma^{\alpha_1, \alpha_2}_{n_1, n_2}(x)|\) is estimated in terms of \(\lambda_j(i_j)\) and \(\log\log n_j\) or \(\log n_j\), \(j= 1,2\), respectively; where \(f(x)\) is the sum of series \((*)\) in the metric of \(L^2(0, 1)\). The rates of almost everywhere summability obtained in three theorems are also shown to be the best possible in the class of all double ONS on \((0,1)\).