  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.
