Journal of computational and applied mathematics vol:133 issue:1-2 pages:635-645

Conference:

5th International Symposium on Orthogonal Polynomials, Special Functions and their Applications location:Patras, Greece date:20-24 September 1999

Abstract:

Let w(t) be a positive weight functionon the interval [-π,π] and associate the positive definite inner product on the unit circle of the complex plane by
<f,g>w=(1/2π)∫_{x=-π...π} f(e^{it})g(e^{it})*w(t)dt.
For a sequence of points {a_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-ã_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(z) 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(z) 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 Szegö in the polynomial case, that is when all a_k = 0.

As an important consequence we find that under the above conditions on the weight w and the points {a_k:k=1...∞}, the Cesàro means of the series sn converge uniformly to the function F in |z| ≤ 1 if the boundary function f(t):=F(e^{it}) is continous on [0,2π]. This can be seen as a generalization of Fejér's Theorem.