Weighted Sobolev orthogonal polynomials on the unit ball

2012/11/12 by Perez, Teresa E., Pinar, Miguel A., Xu, Yuan
#33C50 #42C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.2489

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[∫_\BB^d f(x) g(x) W_μ(x) dx + λ∫_\BB^d ∇ f(x) ⋅ ∇ 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 ⋅,⋅\ra, can be generated through a recursive formula.

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