Bernstein Equiconvergence and Fej Er Type Theorems for General Rational Fourier Series Bernstein Equiconvergence and Fej Er Type Theorems for General Rational Fourier Series

Let w() be a positive weight function on the interval ;) and associate the positive deenite inner product on the unit circle of the complex plane by hf; gi w = 1 2 R f(e ii)g(e ii)w()d. For a sequence of points f k g 1 k=1 included in a compact subset of the open unit disk, we consider the orthogonal rational functions (ORF) f k g 1 k=0 that are obtained by orthogonalization of the sequence f1; z== 1 ; z 2 == 2 ; : : :g where k (z) = Q k j=1 (1 j z), with respect to this inner product. In this paper we prove that s n (z)S n (z) tends to zero in jzj 1 if n tends to 1, where s n is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF f k g 1 k=0 and S n is the nth partial sum of the ordinary power series expansion of F. The main condition on the weight is that it satisses a Lipschitz-Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg} o in the polynomial case, that is when all k = 0. As an important consequence we nd that under the above conditions on the weight w and the points f k g 1 k=1 , the Cess aro means of the series s n converge uniformly to the function F in jzj 1 if the boundary function f() := F (e ii) is continous on 0; 2]. This can be seen as a generalization of Fej er's Theorem.