A note on the Geronimus transformation and Sobolev orthogonal polynomials

Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle

Let L̂ be a positive definite bilinear functional on the unit circle defined on Pn, the space of polynomials of degree at most n. Then its Geronimus transformation L is defined by L̂(p, q) = L ( (z − α)p(z), (z − α)q(z) ) for all p, q ∈ Pn, α ∈ C. Given L̂, there are infinitely many such L which can be described by a complex free parameter. The Hessenberg matrix that appears in the recurrence rela...

Sobolev Orthogonal Polynomials on a Simplex

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function Wγ(x) = x γ1 1 · · ·x γd d (1− |x|)d+1 when all γi > −1 and they are eigenfunctions of a second order partial differential operator Lγ . The singular cases that some, or all, γ1, . . . , γd+1 are −1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of Lγ ...

Old and New Geronimus Type Identities for Real Orthogonal Polynomials

Let be a positive measure on the real line, with orthogonal polynomials fpng and leading coe¢ cients f ng. The Geronimus type identity 1 jIm zj Z 1 1 P (t) jzpn (t) pn 1 (t)j dt = n 1 n Z P (t) d (t) ; valid for all polynomials P of degree 2n 2 has known analogues within the theory of orthogonal rational functions, though apparently unknown in the theory of orthogonal polynomials. We present ne...

Note on Certain Orthogonal Polynomials

A family of Sobolev orthogonal polynomials on the unit ball

A family of orthonormal polynomials on the unit ball B of R with respect to the inner product