Weighted Sobolev orthogonal polynomials on the unit ball

Abstract

For the weight function $W_{\mu }\left(x\right)={(1-{\left|x\right|}^2)}^{\mu }$, $\mu >-1$, $\lambda >0$ and $b_{\mu }$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product

$$〈f,g〉=b_{\mu }[{\int }_{{\mathbb{B}}^d}f\left(x\right)g\left(x\right)W_{\mu }\left(x\right)dx+\lambda {\int }_{{\mathbb{B}}^d}\nabla f\left(x\right)\cdot \nabla g\left(x\right)W_{\mu }\left(x\right)dx]$$

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 $〈\cdot ,\cdot 〉$, can be generated through a recursive formula.