On Generalizations of Fatou's Theorem for the Integrals with General Kernels

We define lambda(r)-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stieltjes integrals for general kernels {phi(r)}, forming an approximation of identity. We prove that the bound lim(r -> 1) sup lambda(r)parallel to phi(r)parallel to(infinity) < infinity is necessary and sufficient for almost everywhere lambda(r)-convergence of the integrals integral(T) phi r(t-x)d mu(t).