Spectral transformations for Hermitian Toeplitz matrices
Journal of Computational and Applied Mathematics
Orthogonal polynomials and measures on the unit circle. The Geronimus transformations
Journal of Computational and Applied Mathematics
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Let $\cal{L}$ be a positive definite bilinear functional, then the Uvarov transformation of $\cal{L}$ is given by $\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1})$ $\overline{q}(\overline{\alpha})$ where $|\alpha| 1, m \in \mathbb{C}$ . In this paper we analyze conditions on m for $\cal{U}$ to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by 驴, n and the moments associated with $\cal{L}$ . We also give an upper bound for the radius of this circle that depends only on 驴 and n. This and other conditions on m are visualized for some examples.