On a spectral theorem in paraorthogonality theory

Abstract

Motivated by the works of Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.

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.…”

“…I n,2 +1 is a Szegő-Peherstorfer q.f. exact in L 2(n− )−1 (ω) with prefixed nodes {α i } 2 i=1 ⊂ T if and only if W n = Q n,2 +1 as in (14) satisfying the hypothesis of Theorem 3.12 where τ is such that τ P (0) − δ n− = 0 and ω = τ P (0)−δ n− τ P (0)−δ n− . 2.…”

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!).…”

“…Indeed some simple examples when n = 1 show that, in general, there exists a one parameter family of choices for (α 0 , α 1 , λ, µ) leading to zeros (w 1 , w 2 , y 1 ). Nevertheless, Castillo et al [11] do have a Wendroff type theorem in the context of (3)! They consider sequences of monic polynomials {Ξ j } ∞ j=0 obeying a three term recurrence relation Ξ n+1 (z) = (z + β n )Ξ n (z) − γ n Ξ n−1 (z) (10.1) with n = 1, 2, .…”

“…with Ξ 0 = 1 and Ξ 1 (z) = z + β 0 for suitable β j ∈ ∂D, j ≥ 0 and γ j ∈ C \ {0}, j ≥ 1. [11] show that, under suitable hypotheses, there exist Verblunsky coefficients {α j } ∞ j=0 ∈ D ∞ and {λ j } ∞ j=0 ∈ ∂D ∞ so that Ξ j are the associated POPUC but, when they exist, {α j } ∞ j=0 ∈ D ∞ and {λ j } ∞ j=0 ∈ ∂D ∞ are not unique but depend on an arbitrary choice of λ 0 ∈ ∂D. They proved that given sets of zeros as in case (3), there is a unique choice of {β j } n j=0 and {γ j } n j=0 and so {Ξ j } n−1 j=0 for which Ξ n and Ξ n+1 have the required zeros.…”