Applicability of Multiplicative Renormalization Method for a Certain Function

We characterize the class of probability measures for which the multiplicative renormalization method can be applied for the function h(x) = 1 √ 1−x to obtain orthogonal polynomials. It turns out that this class consists of only uniform probability measures on intervals and probability measures being supported by one or two points. 1. Multiplicative Renormalization Method Let μ be a probability measure on R with finite moments of all orders. Then we can apply the Gram-Schmidt orthogonalization process to the sequence {x}n=0 to get an orthogonal sequence {Pn(x)}n=0 in the real Hilbert space L(R, μ). Here the leading coefficient of Pn is 1 for each n. It is well known that these polynomials satisfy the recursion formula: (x− αn)Pn(x) = Pn+1(x) + ωn−1Pn−1(x), n ≥ 0, (1.1) where by convention ω−1 = 1 and P−1 = 0. The numbers αn, ωn, n ≥ 0, are known as the Jacobi–Szegö parameters of μ. It is natural to ask whether there is a method for deriving {Pn(x)}n=0 from μ. A method, called multiplicative renormalization method, has been introduced in [3, 4] to answer this question. This method starts with an analytic function h(x) at 0. Then we define two functions