arXiv subjects

Brian Simanek

Publications and source records attributed to Brian Simanek.

41 records · Page 3Linked to original sources

arXiv2012

A New Approach to Ratio Asymptotics for Orthogonal Polynomials

We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is determined only by its leading coefficient and its normalization. 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 - one has ratio asymptotics along a sequence of asymptotic density 1.

math.SP↗

arXiv2011

Asymptotic Properties of Extremal Polynomials Corresponding to Measures Supported on Analytic Regions

Let G be a bounded region with simply connected closure and having analytic boundary and let mu be a positive measure carried by the closure of G together with finitely many pure points outside G. We provide estimates on the norms of the monic polynomials of minimal norm in the space L^q(mu) for q>0. In case the norms converge to 0, we provide estimates on the rate of convergence, generalizing several previous results. Our most powerful result concerns measures mu that are perturbations of measures that are absolutely continuous with respect to the push-forward of a product measure near the boundary of the unit disk. Our results and methods also yield information about the strong asymptotics of the extremal polynomials and some information concerning Christoffel functions.

math.CA↗

arXiv2011

Weak Convergence of CD Kernels: A New Approach on the Circle and Real Line

Let m be a probability measure supported on some infinite and compact set K in the complex plane and let p_n(z) be the corresponding degree n orthonormal polynomial with positive leading coefficient. Let v_n be the normalized zero counting measure for the polynomial p_n and let u_n be the probability measure given by (n+1)u_n=K_n(z,z)m, where K_n(z,w) is the reproducing kernel for polynomials of degree at most n. If m is supported on a compact subset of the real line or the unit circle, we provide a new proof of a 2009 theorem due to Simon, that for any fixed natural number k, the k^{th} moment of u_n and v_{n+1} differ by at most O(1/n) as n tends to infinity.

math.SP↗

arXiv2010

Zeros of non-Baxter paraorthogonal polynomials on the unit circle

We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.

math.SP↗

arXiv2009

Semilocal formal fibers of principal prime ideals

Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide necessary and sufficient conditions for T to be the completion of an integral domain A containing the prime ideal pA whose formal fiber is semilocal with maximal ideals the elements of C.

math.AC↗