Publications of Adhemar Bultheel
Boundary asymptotics for orthogonal rational functions on the unit circle

Boundary asymptotics for orthogonal rational functions on the unit circle


Adhemar Bultheel       Patrick Van gucht


Abstract: Let $w(t)$ be a positive weight function on the unit circle of the complex plane. For a sequence of points $\{\alpha_k\}_{k=1}^\infty$ included in a compact subset of the unit disk, we consider the orthogonal rational functions $\phi_n$ that are obtained by orthogonalization of the sequence $\{1,z/\pi_1,z^2/\pi_2,...\}$ where $\pi_k(z) = \prod_{j=1}^k (1-\overline{\alpha}_jz)$, with respect to the inner product $$ \langle f,g \rangle = \frac{1}{2\pi}\int_{-\pi}^\pi f(e^{it})\overline{g(e^{it})}w(t)dt. $$ We discuss in this paper the behaviour of $\phi_n(t)$ for $|t|=1$ and $n\to\infty$ under certain conditions. 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 Szegő in the polynomial case, that is when all $\alpha_k=0$.

Status:
Published: Acta Applicandae Mathematicae, vol. 61, pages 333-349, 2000

BiBTeX entry: 

   @article{ArtBVG99,
      author = "A. Bultheel and Van gucht, P.",
      title = "Boundary asymptotics for orthogonal rational functions on the unit circle",
      journal = "Acta Applicandae Mathematicae",
      year = "2000",
      url = "http://nalag.cs.kuleuven.be/papers/ade/icra99/index.html",
      volume = "61",
      pages = "333-349",
      DOI = "10.1023/A:1006409205633",
      LIMO = "1128278",
      ZBL = "0969.42012",
      MR = "2001g:42049",
   }
   @unpublished{AbsBVg99,
      author = "A. Bultheel and Van gucht*, P.",
      title = "Boundary asymptotics for orthogonal rational functions on the unit circle",
      note = "Presented at International Conference on Rational Approximation location:University of Antwerp (UIA), Antwerp, BE",
      year = "1999",
      month = "June 6-11",
      url = "http://nalag.cs.kuleuven.be/papers/ade/icra99/index.html",
      LIRIAS = "167226",
   }
File(s): preprint.pdf (230K)

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