Sobolev approximation on the ball
Marriaga Misael E.  1@  , Recarte Marlon J.  2  , Pérez Teresa E.  3  , Piñar Miguel A.  3  
1 : Universidad Rey Juan Carlos [Madrid]
2 : Universidad Nacional Autónoma de Honduras
3 : Universidad de Granada

For $\mu>0$, let
(f,g)_{\mu}=f(0)\,g(0)+\lambda \int_{B^2}\nabla f(x)\cdot \nabla g(x)\,(1-||x||^2)^{\mu}dx, \quad \lambda>0,
be a Sobolev inner product defined on the linear space of polynomials of $d$ variables. Here $\nabla f$ is the gradient of $f$, $B^d$ is the unit ball of $\mathbb{R}^d$ and $||x||$ is the usual Euclidean norm of $x\in \mathbb{R}^d$. In this work, we determine an explicit orthogonal polynomial basis associated with $(\cdot,\cdot)_{\mu}$ and study approximation properties of Fourier expansions in terms of this basis. In particular, we deduce relations between the partial Fourier sums in terms of the Sobolev polynomials and the partial Fourier sums in terms of the classical ball polynomials. We give an estimate of the approximation error by polynomials of degree at most $n$ in the corresponding Sobolev space. We also give upper bounds for the reproducing kernels for the Sobolev projection operators on the space of polynomials of degree at most n.

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