27 research outputs found
Zeros of orthogonal polynomials on the real line
Let be orthogonal polynomials associated to a measure of
compact support in . If , we show there is a
so that for all , either or has no zeros in . If is an isolated point of , we show there is a
so that for all , either or has at most one zero in
. We provide an example where the zeros of are
dense in a gap of .Comment: (preliminary version
Zeros of orthogonal polynomials on the real line
Orientadores: Dimitar Kolev Dimitrov, Roberto AndreaniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho são obtidos resultados sobre o comportamento de zeros de polinômios ortogonais. Sabe-se que todos eles são reais e distintos e fazem papel importante de nós das mais utilizadas fórmulas de integração numérica, que são as fórmulas de quadratura de Gauss. São obtidos resultados sobre a localização e a monotonicidade dos zeros, considerados como funções dos correspondentes parâmetros, dos polinômios ortogonais clássicos. Apresentaremos também vários resultados que tratam da localização, monotonicidade e da assintótica de zeros de certas classes de polinômios ortogonais relacionados com as medidas clássicasAbstract: Results concerning the behaviour of zeros of orthogonal polynomials are obtained. It is known that they are real and distinct and play as important role as node of the most frequently used rules for numerical integration, the Gaussian quadrature formulae. Result about the location and monotonicity of the zeros, considered as functions of parameters involved in the measure, are provided. We present various results that treat questions about location, monotonicity and asymptotics of zeros of certain classes of orthogonal polynomials with respect to measure that are closely related to the classical onesDoutoradoAnalise AplicadaDoutor em Matemática Aplicad
Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies a positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs
Fine structure of the zeros of orthogonal polynomials III: Periodic recursion coefficients
We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them
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
First and second kind paraorthogonal polynomials and their zeros
Given a probability measure with infinite support on the unit circle
, we consider a sequence of paraorthogonal
polynomials \h_n(z,\lambda) vanishing at where \lambda \in \T
is fixed. We prove that for any fixed z_0 \not \in \supp(d\mu) distinct from
, we can find an explicit independent of such that either
\h_n or \h_{n+1} (or both) has no zero inside the disk , with
the possible exception of .
Then we introduce paraorthogonal polynomials of the second kind, denoted
\s_n(z,\lambda). We prove three results concerning \s_n and \h_n. First,
we prove that zeros of \s_n and \h_n interlace. Second, for an
isolated point in \supp(d\mu), we find an explicit radius \rt such that
either \s_n or \s_{n+1} (or both) have no zeros inside B(z_0,\rt).
Finally we prove that for such we can find an explicit radius such that
either \h_n or \h_{n+1} (or both) has at most one zero inside the ball
B(z_0,\rt).Comment: To appear in the Journal of Approximation Theor