A connection between Szegő–Lobatto and quasi Gauss-type quadrature formulas

Zeros of quasi-paraorthogonal polynomials and positive quadrature

“…The third part described in (18) is the classical Schur-Cohn test (see [43,17]) for the polynomial P ℓ .…”

“…Since then, they have been widely studied, not only because of their own interest [45], but also in many applications such as the trigonometric moment problem [1], complex approximation [49], probability and statistics [32], prediction theory [50], systems theory, networks, circuits and scattering [24], signal processing [21], and many more, but also because of their intimate connection with OPRL (see e.g. [8,18]). However, these polynomials present important differences with respect to OPRL, in particular, concerning the above properties, since their zeros are located outside of the support of the measure.…”

Zeros of quasi-paraorthogonal polynomials and positive quadrature

“…The third part described in (18) is the classical Schur-Cohn test (see [43,17]) for the polynomial P . Remark 3.13.…”

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“…The construction of Szegő-Lobatto quadrature formulas also requires the computation of an n-th POPUC with a specific value of the parameter τ n ; see [Cruz-Barroso et al 2015].…”

“…The counterpart to the deficiency in the orthogonality conditions for POPUC, which are not orthogonal to the constants, is the fact that for a given measure and a fixed n, the POPUC of degree n is not unique, and basically depends on one unimodular free parameter. Equivalently, in quadrature terminology, we have a one-parameter family of n-point SQ formulas, exact in a subspace of Laurent polynomials of dimension 2n − 1, instead of 2n; see, e.g., [Cruz-Barroso et al 2007;Peherstorfer 2011].…”