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arXiv · 1211.2489

Weighted Sobolev orthogonal polynomials on the unit ball

Teresa E. Perez·Miguel A. Pinar·Yuan Xu
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Abstract

For the weight function $W_μ(x) = (1-|x|^2)^μ$, $μ> -1$, $λ> 0$ and $b_μ$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_μ[\int_{\BB^d} f(x) g(x) W_μ(x) dx + λ\int_{\BB^d} \nabla f(x) \cdot \nabla g(x) W_μ(x) dx]} $$ 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.

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Teresa E. Perez, Miguel A. Pinar, Yuan Xu. 2012-11-12. Weighted Sobolev orthogonal polynomials on the unit ball. https://arxiv.org/abs/1211.2489

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