Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

Abstract

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Zeros of quasi-paraorthogonal polynomials and positive quadrature

“…Invariant polynomials, orthogonal to this subspace for = 0 were coined paraorthogonal as they were introduced in [34], (see also [28, Theorem III]), and they were later studied by many [22,23,12,13,15,14,31,51,36,38]. We shall refer to them as quasi-paraorthogonal polynomials (QPOPUC) of order 1.…”

“…This problem has a matrix interpretation, that extends the case = 1. Let us assume that there exists a unique solution P for the system (15) and that it is given by P…”

Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators

“…The definition goes back to Delsarte-Genin [19] and Jones et. al [45]; among later papers, we mention [7,9,10,11,18,34,58,63,75]. One can show that the n + 1 point measure, dν λ , whose first n Verblunsky coefficients are the first n α j and with α n = λ has Φ n+1 (z; λ) as its n+1st monic OPUC (which has norm 0!).…”

On a spectral theorem in paraorthogonality theory

“…It is known that an arbitrary polynomial with simple zeros on ‫ބ∂‬ is a POPUC with respect to some nontrivial probability measure supported on ‫ބ∂‬ [Castillo et al 2015]. Since n (β n ) = 0 and n+1 (β n ) = 0, and we are interested in the zeros, there is no loss of generality if we assume that…”