Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle

Abstract

It is known that given a pair of real sequences {{cn}n=1 ∞,{dn}n=1 ∞}, with {dn}n=1 ∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients {αn}n=0 ∞ are given by the relationαn−1=ρ‾n−1[[formula presented]],n≥1 where ρ0=1, ρn=∏k=1 n(1−ick)/(1+ick), n≥1 and {mn}n=0 ∞ is the minimal parameter sequence of {dn}n=1 ∞. In this paper we consider the space, denoted by Np, of all nontrivial probability measures such that the associated real sequences {cn}n=1 ∞ and {mn}n=1 ∞ are periodic with period p, for p∈N. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gp between the metric subspaces Np and Vp, where Vp denotes the space of nontrivial probability measures with associated p-periodic Verblunsky coefficients. Moreover, it is shown that the set Fp of fixed points of gp is exactly Vp∩Np and this set is characterized by a (p−1)-dimensional submanifold of Rp. We also prove that the study of probability measures in Np is equivalent to the study of probability measures in Vp. Furthermore, it is shown that the pure points of measures in Np are, in fact, zeros of associated para-orthogonal polynomials of degree p. We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences {cn}n=1 ∞ and {mn}n=1 ∞ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.