37 research outputs found
Szego polynomials: a view from the Riemann-Hilbert window
This is an expanded version of the talk given at the conference
``Constructive Functions Tech-04''. We survey some recent results on canonical
representation and asymptotic behavior of polynomials orthogonal on the unit
circle with respect to an analytic weight. These results are obtained using the
steepest descent method based on the Riemann-Hilbert characterization of these
polynomials.Comment: 23 pages, 7 figures; To appear in Elect. Trans. Num. Anal.
Apparently, due to some missing .sty file, all integrals in version 2 were
gone. Version 3 is a mere correction of this proble
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes
polynomials - polynomial solutions of a second order differential equations
with complex polynomial coefficients. In the case when all zeros of the leading
coefficients are all real, zeros of the Heine-Stieltjes polynomials were
interpreted by Stieltjes as discrete distributions minimizing an energy
functional. In a general complex situation one deals instead with a critical
point of the energy. We introduce the notion of discrete and continuous
critical measures (saddle points of the weighted logarithmic energy on the
plane), and prove that a weak-* limit of a sequence of discrete critical
measures is a continuous critical measure. Thus, the limit zero distributions
of the Heine-Stieltjes polynomials are given by continuous critical measures.
We give a detailed description of such measures, showing their connections with
quadratic differentials. In doing that, we obtain some results on the global
structure of rational quadratic differentials on the Riemann sphere that have
an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math.
Physic