Numerical Algorithms

, Volume 59, Issue 1, pp 51–62 | Cite as

When is the Uvarov transformation positive definite?

Original Paper


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.


Linear functional Spectral transformation Toeplitz matrix Positive definite 

Mathematics Subject Classifications (2010)

30E05 15A63 


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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenHeverleeBelgium

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