16 research outputs found
Cantor polynomials and some related classes of OPRL
We explore the spectral theory of the orthogonal polynomials associated to the classical Cantor measure and similar singular continuous measures. We prove regularity in the sense of Stahl–Totik with polynomial bounds on the transfer matrix. We present numerical evidence that the Jacobi parameters for this problem are asymptotically almost periodic and discuss the possible meaning of the isospectral torus and the Szegő class in this context
Equilibrium measures and capacities in spectral theory
This is a comprehensive review of the uses of potential theory in studying
the spectral theory of orthogonal polynomials. Much of the article focuses on
the Stahl-Totik theory of regular measures, especially the case of OPRL and
OPUC. Links are made to the study of ergodic Schrodinger operators where one of
our new results implies that, in complete generality, the spectral measure is
supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in
the region of strictly positive Lyapunov exponent. There are many examples and
some new conjectures and indications of new research directions. Included are
appendices on potential theory and on Fekete-Szego theory
Orthogonal polynomials for the weakly equilibrium Cantor sets
Let be the weakly equilibrium Cantor type set introduced in [10].
It is proven that the monic orthogonal polynomials with respect to
the equilibrium measure of coincide with the Chebyshev polynomials
of the set. Procedures are suggested to find of all degrees and the
corresponding Jacobi parameters. It is shown that the sequence of the Widom
factors is bounded below
Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings
Let be a probability measure with an infinite compact support on
. Let us further assume that is a sequence of
orthogonal polynomials for where is a sequence of
nonlinear polynomials and for all
. We prove that if there is an such that
is a root of for each then the distance between any two
zeros of an orthogonal polynomial for of a given degree greater than
has a lower bound in terms of the distance between the set of critical points
and the set of zeros of some . Using this, we find sharp bounds from below
and above for the infimum of distances between the consecutive zeros of
orthogonal polynomials for singular continuous measures.Comment: Contains less typo
Right limits and reflectionless measures for CMV matrices
We study CMV matrices by focusing on their right-limit sets. We prove a CMV
version of a recent result of Remling dealing with the implications of the
existence of absolutely continuous spectrum, and we study some of its
consequences. We further demonstrate the usefulness of right limits in the
study of weak asymptotic convergence of spectral measures and ratio asymptotics
for orthogonal polynomials by extending and refining earlier results of
Khrushchev. To demonstrate the analogy with the Jacobi case, we recover
corresponding previous results of Simon using the same approach
Fast and accurate computation of the logarithmic capacity of compact sets
We present a numerical method for computing the logarithmic capacity of
compact subsets of , which are bounded by Jordan curves and have
finitely connected complement. The subsets may have several components and need
not have any special symmetry. The method relies on the conformal map onto
lemniscatic domains and, computationally, on the solution of a boundary
integral equation with the Neumann kernel. Our numerical examples indicate that
the method is fast and accurate. We apply it to give an estimate of the
logarithmic capacity of the Cantor middle third set and generalizations of it