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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:133

Article; Proceedings Paper

Van Gucht, P ; Bultheel, A

关键词: equiconvergence;    Fejer theorem;    orthogonal rational functions;   

DOI  :  10.1016/S0377-0427(00)00662-2

来源: Elsevier

【 摘 要 】

Let w(theta) be a positive weight function on the interval [-pi, pi] and associate the positive-definite inner product on the unit circle of the complex plane by (w) = 1/2 pi integral (pi)(-pi) F(e(i theta))<(G(e(i)))over bar>w(theta )d theta For a sequence of points {alpha (k)}(k=1)(infinity) included in a compact subset of the open unit disk, we consider the orthogonal rational functions (ORF) {phi (k)}(k=0)(infinity) that are obtained by orthogonalization of the sequence {1, z/pi (1),z(2)/pi (2),...} where pi (k)(z)=Pi (k)(i=1)(1 - <()over bar>(j)z), with respect to this inner product. In this paper we prove that s(n)(z) - S-n(z) tends to zero in \z \ less than or equal to 1 if n tends to infinity, where s(n) is the nth partial sum of the expansion of a bounded analytic function F in terms of the ORF {phi (k)}(k=0)(infinity) 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 Szego in the polynomial case, that is when all alpha (k) = 0. As an important consequence we find that under the above conditions on the weight w and the points {alpha (k)}(k=1)(infinity), the Cesaro means of the series s(n) converge uniformly to the function F in \z \ less than or equal to 1 if the boundary function f(theta):=F(e(i theta)) is continuous on [0,2 pi]. This can be seen as a generalization of Fejer's Theorem. (C) 2001 Elsevier Science B.V. All rights reserved.

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