2016/08/19 by Cleonice F. Bracciali, Jairo Silva, Bracciali, Cleonice F. +5
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1608.08079
openalex publication_date 2016/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It was shown recently that associated with a pair of real sequences\n cn n=1\∞, dn n=1\∞ , with\n dn n=1\∞ a positive chain sequence, there exists a unique\nnontrivial probability measure \μ on the unit circle. The Verblunsky\ncoefficients \αn n=0\∞ associated with the orthogonal\npolynomials with respect to \μ are given by the relation \n
alphan-1=
overline
taun-1
left[
frac1-2mn-icn1-icn
right],\n
quad n
geq 1, where \τ0 = 1,\n\τn=\∏k=1n(1-ick)/(1+ick), n \≥ 1 and\n mn n=0\∞ is the minimal parameter sequence of\n dn n=1\∞.\n In this manuscript we consider this relation and its consequences by imposing\nsome restrictions of sign and periodicity on the sequences\n cn n=1\∞ and mn n=1\∞. When the sequence \n cn n=1\∞ is of alternating sign, we use information about\nthe zeros of associated para-orthogonal polynomials to show that there is a gap\nin the support of the measure in the neighbourhood of z= -1. Furthermore, we\nshow that it is possible to ge -nerate periodic Verblunsky coefficients by\nchoosing periodic sequences cn n=1\∞ and\n mn n=1\∞ with the additional restriction c2n=-c2n-1,\n , n\≥ 1. We also give some results on periodic Verblunsky coefficients from\nthe point of view of positive chain sequences. An example is provided to\nillustrate the results obtained.\n