Sobolev orthogonal polynomials defined via gradient on the unit ball

Details Sobolev orthogonal polynomials defined via gradient on the unit ball Sobolev orthogonal polynomials defined via gradient on the unit ball

2006-12-18

arxiv.org/abs/math/0612527v1

Mathematics

Classical Analysis and ODEs

12 pages

Scientific paper

An explicit family of polynomials on the unit ball $B^d$ of $\RR^d$ is constructed, so that it is an orthonormal family with respect to the inner product $$ < f,g > = \rho \int_{B^d}\nabla f(x)\cdot \nabla g(x) dx + \CL (fg), $$ where $\rho >0$, $\nabla$ is the gradient, and $\CL(fg)$ is either the inner product on the sphere $S^{d-1}$ or $f(0)g(0)$.

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