Rate of approximation by rectangular partial sums of double orthogonal series (Q1375622)

The author studies the rate of almost everywhere approximation to square integrable functions by the rectangular partial sums of their orthogonal expansion \[ \sum_{j=0}^\infty \sum_{k=0}^\infty a_{jk}\phi_{jk}(x), \] where \(\{\phi_{jk}(x)\}\) is a double orthonormal system on a positive measure space. He obtains best possible rates of approximation under appropriate conditions imposed on the decrease of the coefficients \(\{a_{jk}\}\), provided the class \(\Phi\) of all double orthogonal systems is taken into consideration. He also shows in examples that the conditions imposed on \(\{a_{jk}\}\) are necessary in order to guarantee the rate in question on the whole class \(\Phi\). These results generalize and extend recent theorems achieved by K. Tandori, V.I. Koljada, H. Schwinn, and the reviewer.