The point-wise convergence of general rational Fourier series

Let {ak}k=18 be a sequence included in a compact subset of the unit disk D; we consider the rational Fourier series {fk(eix)}k=08 that are obtained by orthogonalization of the Blaschke product sequence {B0(eix)?=?1,B1(eix),????,Bn(eix),????}, where Bn(eix)=?k=1neix-ak/1-akeix. In order to study the point-wise convergence of this rational Fourier series, let the partial sums of f???L1[?-?p,p] be defined?as Sn(f)(x):=?k=-nn?f,fk?fk(eix), where f-k(eix)=fk(eix) for k?N. In this paper, we will show that the conditions for the point-wise convergence of f(x)?-?Sn(f)(x) is the same as that of the Fourier series. More precisely, one is the DirichletDini criterion, and the other is the Jordan test. Copyright (c)?2012 John Wiley & Sons, Ltd.