| Full text | |
| Author(s): |
Bracciali, Cleonice F.
;
Silva, Jairo S.
;
Sri Ranga, A.
;
Veronese, Daniel O.
Total Authors: 4
|
| Document type: | Journal article |
| Source: | Journal of Mathematical Analysis and Applications; v. 445, n. 1, p. 719-745, JAN 1 2017. |
| Web of Science Citations: | 0 |
| Abstract | |
It is known that given a pair of real sequences [[cn](n=1)(infinity), [d(n)](n=1)(infinity)], with [d(n)](n=1)(infinity) a positive chain sequence, we can associate a unique nontrivial probability measure mu on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients [alpha(n))(n=0)(infinity) are given by the relation alpha(n-1) = (rho) over bar (n-1) {[}1 - 2m(n) - ic(n)/1 - ic(n)], n >= 1, where rho(0) = 1, rho(n) = Pi(n)(k=1)(1 - ic(k))/ (1 + ic(k)), n >= 1 and [m(n)](n=0)(infinity) is the minimal parameter sequence of [d(n)](n=1)(infinity). In this paper we consider the space, denoted by N-p, of all nontrivial probability measures such that the associated real sequences [c(n)](n=1)(infinity) and [m(n)](n=1)(infinity) are periodic with period p, for p is an element of 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 g(p) between the metric subspaces N-p and V-p, where V-p denotes the space of nontrivial probability measures with associated p-periodic Verblunsky coefficients. Moreover, it is shown that the set F-p of fixed points of g(p) is exactly V-p boolean AND N-p, and this set is characterized by a (p - 1)-dimensional submanifold of R-p. We also prove that the study of probability measures in N-p is equivalent to the study of probability measures in V-p. Furthermore, it is shown that the pure points of measures in N-p 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 [c(n)](n=1)(infinity) and [m(n)](n=1)(infinity) are limit periodic with period p. Finally, we give some examples to illustrate the results obtained. (C) 2016 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 14/22571-2 - Orthogonal and Similar Polynomials with some analytical and numerical applications |
| Grantee: | Cleonice Fátima Bracciali |
| Support Opportunities: | Regular Research Grants |