Laurent polynomial perturbations of linear functionals. An inverse problem

Kenier Castillo, Luis Garza, and Francisco Marcellán

Abstract

Given a linear functional $\mathcal{L}$ in the linear space ${\bf P}$ of polynomials with complex coefficients, we analyze those linear functionals $\tilde{\mathcal{L}}$ such that, for a fixed $\alpha\in{\bf C}$, $\langle\tilde{\mathcal{L}},(z+z^{-1}-(\alpha+\bar\alpha))p\rangle = \langle \mathcal{L},p \rangle$ for every $p\in{\bf P}$. We obtain the relation between the corresponding Carathéodory functions in such a way that a linear spectral transform appears. If $\mathcal{L}$ is a positive definite linear functional, the necessary and sufficient conditions in order for $\tilde{\mathcal{L}}$ to be a quasi-definite linear functional are given. The relation between the corresponding sequences of monic orthogonal polynomials is presented.

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Key words

Orthogonal polynomials, linear functionals, Laurent polynomials, linear spectral transformations.

42C05.

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