Almost Everywhere Convergence of Orthogonal Series Revisited

We deal with single and double orthogonal series and give sufficient conditions which ensure their convergence almost everywhere. Among others, we prove that if ∑∞j = 3 ∑∞k = 3a2jk log j log k log2+ (1/a2jk) < ∞, then the series ∑j ∑kajkψjk(x) converges a.e. in Pringsheim′s sense for each double orthonormal system {ψjk(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.