When is the Uvarov transformation positive definite?

Let L be a positive definite bilinear functional, then the Uvarov transformation of L is given by U(p, q)= L(p, q) + m p(alpha)(q) over bar(alpha(-1)) + (m) over bar p(alpha(-1)) (q) over bar((alpha) over bar) where |a|>1, m is an element of C. In this paper we analyze conditions on m for 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 alpha, n and the moments associated with L. We also give an upper bound for the radius of this circle that depends only on alpha and n. This and other conditions on m are visualized for some examples.