Best polynomial approximation on the unit ball

Let E-n (f )(mu) be the error of best approximation by polynomials of degree at most n in the space L-2 (pi(mu), B-d), where B-d is the unit ball in R-d and pi(mu)(x) = (1 -parallel to x parallel to(2))(mu) mu > -1. Our main result shows that, for s is an element of N. E-n(f)(mu) <= cn(-2s)[En-2s(Delta(s)f)(mu+2s) + E-n(Delta(s)(0)f)(mu)]. where Delta and Delta(0) are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right-hand side contains odd-order derivatives.