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

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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= 1 2 π ∫ − π π F( e iθ ) G( e iθ ) w(θ) dθ. For a sequence of points { α k } k=1 ∞ included in a compact subset of the open unit disk, we consider the orthogonal rational functions ( ORF) { φ k } k=0 ∞ that are obtained by orthogonalization of the sequence {1, z/π 1, z 2/π 2,…} where π k(z)= ∏ j=1 k (1− α ̄ jz) , with respect to this inner product. In this paper we prove that s n(z)− S n(z) tends to zero in | z|⩽1 if n tends to ∞, where s n is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF { φ k } 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 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 α k =0. As an important consequence we find that under the above conditions on the weight w and the points { α k } k=1 ∞, the Cesàro means of the series s n converge uniformly to the function F in | z|⩽1 if the boundary function f(θ) ≔ F( e iθ ) is continuous on [0,2π]. This can be seen as a generalization of Fejér's Theorem.