When is the Uvarov transformation positive definite?

Abstract

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 α, n and the moments associated with $\cal{L}$. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.