Linear spectral transformation and Laurent polynomials
Let \(\mathcal{L}\) be Hermitian linear functional on the space of Laurent polynomials \(\Lambda=\,\)span\(\{z^n\}_{n\in\mathbb{Z}}\) and let \(P_n(z)\) a complex monic polynomial. Let define the new functional \(\mathcal{L}_I: \, \langle \mathcal{L}_I,q\rangle=\langle \mathcal{L},(P_n(z)-\overline{P}_n(1/z))/(2i)\,q\rangle\) and \(\mathcal{L}_R: \, \langle \mathcal{L}_R,q\rangle=\langle \mathcal{L},(P_n(z)+\overline{P}_n(1/z))/2\,q\rangle\). In this paper the authors study the relation between the Hessenberg matrices corresponding to the functional \(\mathcal{L}\) and \(\mathcal{L}_R\) as well as the relation between the Verblunsky coefficients for the polynomials associated with \(\mathcal{L}_R\) and \(\mathcal{L}\). Finally some representative examples are studied in details.
orthogonal polynomials on the unit circle
scientific article; zbMATH DE number 5635332
scientific article