Spectral approximation on the unit ball

Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces and explicit construction of approximating polynomials. One major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the Sobolev space $W_2^s(\ball)$. As an application, a direct and efficient spectral-Galerkin method based on our orthogonal polynomials is proposed for the second and the fourth order elliptic equations on the unit ball, its optimal error estimates are explicitly derived for both procedures in the Sobolev spaces and, finally, numerical examples are presented to illustrate the theoretic results.