Weighted Sobolev orthogonal polynomials on the unit ball

Abstract

For the weight function Wμ(x)=(1−|x|2)μ, μ>−1, λ>0 and bμ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product 〈f,g〉=bμ[∫Bdf(x)g(x)Wμ(x)dx+λ∫Bd∇f(x)⋅∇g(x)Wμ(x)dx] are constructed in terms of spherical harmonics and a sequence of Sobolev orthogonal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to 〈⋅,⋅〉, can be generated through a recursive formula.