Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure (original) (raw)
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It was shown recently that associated with a pair of real sequences {{c n } ∞ n=1 , {d n } ∞ n=1 }, with {d n } ∞ n=1 a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {α n } ∞ n=0 associated with the orthogonal polynomials with respect to μ are given by the relation α n−1 = τ n−1 1 − 2m n − ic n 1 − ic n , n ≥ 1, where τ 0 = 1, τ n = n k=1 (1 − ic k)/(1 + ic k), n ≥ 1 and {m n } ∞ n=0 is the minimal parameter sequence of {d n } ∞ n=1. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {c n } ∞ n=1 and {m n } ∞ n=1. When the sequence {c n } ∞ n=1 is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the Communicated by Antonio José Silva Neto.