Zeros of quasi-paraorthogonal polynomials and positive quadrature

Positive quadrature formula on an arc of the unit circle

“…In general, we can only guarantee that Bn has n − 1 of its zeros in (a, b), see Theorem 2.3. However, by [1,Theorem 3.4…”

“…|δ|+ηn . That can be derived from [1] where the relation between the invariance parameter Bn(0) and the orthogonality parameter τn of a general quasi-paraorthogonal polynomial Bn is given. We will give a simplified derivation in Lemma A.3.…”

“…The authors in [1,2] introduced a nested sequence of subspaces of Laurent polynomials that allows us to obtain properties analogous to those on the real line for the quadrature formulas on the unit circle. In addition they introduce the quasi-paraorthogonal polynomials as nodal polynomials to provide positive quadrature rules with maximal domain of exactness.…”

“…Thus, an important aim of this paper is to set a first step in the analysis and construction of quadrature rules on an arc of the unit circle. This paper is the third in this context of paraorthogonality, see [1,2]. The paper is organized as follows.…”

“…This theorem says that the nodal polynomial W n is a quasi-paraorthogonal polynomial of order 2ℓ+1 (see [1]). The paraorthogonal polynomials correspond to ℓ = 0.…”