  Generalized translation operators for orthogonal expansions with respect to
families of weight functions on the unit ball and on the standard simplex are
studied. They are used to define convolution structures and modulus of
smoothness for these regions, which are in turn used to characterize the best
approximation by polynomials in the weighted $L^p$ spaces. In one variable,
this becomes the generalized translation operator for the Gegenbauer polynomial
expansions.
