16 research outputs found
On the leading coefficient of polynomials orthogonal over domains with corners
Let be the interior domain of a piecewise analytic Jordan curve without
cusps. Let be the sequence of polynomials that are
orthonormal over with respect to the area measure, with each having
leading coefficient . N. Stylianopoulos has recently proven that
the asymptotic behavior of as is given by where
as and is the reciprocal of the
logarithmic capacity of the boundary . In this paper, we prove that
the estimate for the error term is, in general, best
possible, by exhibiting an example for which The proof makes use of the Faber
polynomials, about which a conjecture is formulated.Comment: 7 page
Asymptotics of polynomials orthogonal over circular multiply connected domains
Let be a domain obtained by removing, out of the unit disk ,
finitely many mutually disjoint closed disks, and for each integer ,
let be the monic th-degree polynomial satisfying the
planar orthogonality condition , . Under a certain assumption on the domain , we establish asymptotic
expansions and formulae that describe the behavior of as
at every point of the complex plane. We also give an asymptotic expansion
for the squared norm
Asymptotic behavior and zero distribution of Carleman orthogonal polynomials
Let be an analytic Jordan curve and let be the
sequence of polynomials that are orthonormal with respect to the area measure
over the interior of . A well-known result of Carleman states that
\label{eq12}
\lim_{n\to\infty}\frac{p_n(z)}{\sqrt{(n+1)/\pi} [\phi(z)]^{n}}= \phi'(z)
locally uniformly on certain open neighborhood of the closed exterior of ,
where is the canonical conformal map of the exterior of onto the
exterior of the unit circle. In this paper we extend the validity of
(\ref{eq12}) to a maximal open set, every boundary point of which is an
accumulation point of the zeros of the 's. Some consequences on the
limiting distribution of the zeros are discussed, and the results are
illustrated with two concrete examples and numerical computations.Comment: 23 pages, 4 figure
A representation for the reproducing kernel of a weighted Bergman space
For a weight function in the unit disk which is the modulus of a finite
product of powers of Blaschke factors, we give a canonical representation for
the reproducing kernel of the corresponding weighted Bergman space in terms of
the values of the kernel and its derivatives at the origin. This yields a
formula for the contractive zero divisor of a Bergman space corresponding to a
finite zero set
Universality for conditional measures of the sine point process
The sine process is a rigid point process on the real line, which means that
for almost all configurations , the number of points in an interval is determined by the points of outside of . In addition, the
points in are an orthogonal polynomial ensemble on with a weight
function that is determined by the points in . We prove a
universality result that in particular implies that the correlation kernel of
the orthogonal polynomial ensemble tends to the sine kernel as the length
tends to infinity, thereby answering a question posed by A.I. Bufetov.Comment: 26 pages, no figures, revised version with Appendix
Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle
For the Riesz and logarithmic potentials, we consider greedy energy sequences
on the unit circle , constructed in such a way that
for every , the discrete potential generated by the first points
of the sequence attains its minimum (say ) at .
We obtain asymptotic formulae that describe the behavior of as
, in terms of certain bounded arithmetic functions with a doubling
periodicity property. As previously shown in \cite{LopMc2}, after properly
translating and scaling , one obtains a new sequence that is
bounded and divergent. We find the exact value of (the value of
was already given in \cite{LopMc2}), and show that the interval
comprises all the limit points of the sequence
.Comment: 20 page
An expansion for polynomials orthogonal over an analytic Jordan curve
We consider polynomials that are orthogonal over an analytic Jordan curve L
with respect to a positive analytic weight, and show that each such polynomial
of sufficiently large degree can be expanded in a series of certain integral
transforms that converges uniformly in the whole complex plane. This expansion
yields, in particular and simultaneously, Szego's classical strong asymptotic
formula and a new integral representation for the polynomials inside L. We
further exploit such a representation to derive finer asymptotic results for
weights having finitely many singularities (all of algebraic type) on a thin
neighborhood of the orthogonality curve. Our results are a generalization of
those previously obtained in [7] for the case of L being the unit circle.Comment: 15 pages, 1 figur