Ratio Asymptotics For Orthogonal Polynomials Research Articles

Stability of ratio asymptotics of orthogonal polynomials under some class of measure perturbations

Stability of ratio asymptotics of orthogonal polynomials under some class of measure perturbations

A new approach to ratio asymptotics for orthogonal polynomials

We use a non-linear characterization of orthonormal polynomials due to Saff in order to prove an equivalence of norm asymptotics and strong asymptotics for polynomials. Several applications of this equivalence are also discussed. One of our main results is that for regular measures on the closed unit disk - including, but not limited to the unit circle - or the interval [-2,2], one has ratio asymptotics along a sequence of asymptotic density 1.

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.

Asymptotics of derivatives of orthogonal polynomials on the unit circle

We show that ratio asymptotics of orthogonal polynomials on the circle imply ratio asymptotics for all their derivatives. Moreover, by reworking ideas of Nevai, we show that uniform asymptotics for orthogonal polynomials on an arc of the unit circle imply asymptotics for all their derivatives.

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

Let μ be a finite positive Borel measure with compact support consisting of an interval [ c , d ] ⊂ R plus a set of isolated points in R ⧹ [ c , d ] , such that μ ′ > 0 almost everywhere on [ c , d ] . Let { w 2 n } , n ∈ Z + , be a sequence of polynomials, deg w 2 n ⩽ 2 n , with real coefficients whose zeros lie outside the smallest interval containing the support of μ . We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form d μ / w 2 n . In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.

Ratio asymptotics of Hermite-Padé polynomials for Nikishin systems

The existence of ratio asymptotics is proved for a sequence of multiple orthogonal polynomials with orthogonality relations distributed among a system of finite Borel measures with support on a bounded interval of the real line which form a so-called Nikishin system. For this result reduces to Rakhmanov's celebrated theorem on the ratio asymptotics for orthogonal polynomials on the real line.

On ratio asymptotics for general polynomials

In this paper some characterizations of the ratio asymptotics for general polynomials are given. These results are extensions and improvements of the ratio asymptotics for orthogonal polynomials and are applicable to the ratio asymptotics for polynomials with disturbed nodes.

New Characterizations of Ratio Asymptotics for Orthogonal Polynomials

In this paper some new characterizations of ratio asymptotics for orthogonal polynomials are given.

Ratio Asymptotics for Polynomials Orthogonal on Arcs of the Unit Circle

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of limn |Φn(0)| and {limn [Φn+1(0)/Φn(0)] , where \( \{\Phi_n(0)\}_{n \geq 0} \) denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered.