Bernstein equiconvergence and Fejér-type theorems for general rational Fourier series

摘要

Let w(θ) be a positive weight function on the interval [−π,π] and associate the positive-definite inner product on the unit circle of the complex plane by〈F,G〉w=12π∫−ππF(eiθ)G(eiθ)w(θ)dθ.For a sequence of points {αk}k=1∞ included in a compact subset of the open unit disk, we consider the orthogonal rational functions (ORF) {φk}k=0∞ that are obtained by orthogonalization of the sequence {1,z/π1,z2/π2,…} where πk(z)=∏j=1k(1−ᾱjz), with respect to this inner product. In this paper we prove that sn(z)−Sn(z) tends to zero in |z|⩽1 if n tends to ∞, where sn is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF {φk}k=0∞ and Sn is the nth partial sum of the ordinary power series expansion of F. The main condition on the weight is that it satisfies a Dini–Lipschitz condition and that it is bounded away from zero. This generalizes a theorem given by Szegő in the polynomial case, that is when all αk=0. As an important consequence we find that under the above conditions on the weight w and the points {αk}k=1∞, the Cesàro means of the series sn converge uniformly to the function F in |z|⩽1 if the boundary function f(θ)≔F(eiθ) is continuous on [0,2π]. This can be seen as a generalization of Fejér's Theorem.