Almost everywhere convergence of orthogonal series revisited (Q1323907)

We deal with single and double orthogonal series and give sufficient conditions which ensure their convergence almost everywhere. Among others, we prove that if \[ \sum^ \infty_{j= 3} \sum^ \infty_{k= 3} a_{jk}^ 2\log j\log k\log^ 2_ +(1/a^ 2_{jk})<\infty, \] then the series \(\sum_ j\sum_ k a_{jk} \psi_{jk}(x)\) converges a.e. in Pringsheim's sense for each double orthonormal system \(\{\psi_{jk}(x)\}\). The interrelation between the well-known Rademacher-Menshov (type) theorems and ours are discussed in detail. At the end, we raise three problems concerning the characterization of a.e. convergence of orthogonal series.