Weighted Sobolev orthogonal polynomials on the unit ball

Abstract

For the weight function $W_\mu(x) = (1-|x|^2)^\mu$, $\mu > -1$, $\lambda > 0$ and $b_\mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_\mu [\int_{\BB^d} f(x) g(x) W_\mu(x) dx + \lambda \int_{\BB^d} \nabla f(x) \cdot \nabla g(x) W_\mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $\la \cdot,\cdot\ra$, can be generated through a recursive formula.