Home > Publications > Reports > Numerical Analysis and Applied Mathematics (TW)


TW 291

P. Van gucht and A. Bultheel
Bernstein equiconvergence and Fejér type theorems for general rational Fourier series

Abstract

Let w(ζ) be a positive weight function on the interval [-&pi,π) and associate the complex definite inner product

⟨f,g⟩ =
1
2π
π
∫
-π
 f(eiθ)
g(eiθ )
 
 w(θ)dθ.
For a sequence of points {αk}k=1 ∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions (ORF) {φn(z)}k=1 ∞ that are obtained by orthogonalization of the sequence {1,z/π1,z2/ π2,...} where πk(z)= ∏j=1k (1-α*jz), 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(z) is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF {φn(z)}k=1 ∞ and Sn(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 Lipschitz-Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szegö 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 on 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(ej θ) is continuous on [0,2 π]. This can be seen as a generalization of Fejèr's theorem.

report.pdf / mailto: P. Van gucht