Best Polynomial Approximation on the Unit Sphere and the Unit Ball

This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify a correct gadget that characterizes smoothness of functions, either a modulus of smoothness or a $K$- functional, the two of which are often equivalent. We will concentrate on characterization of best approximations, given in terms of direct and converse theorems, and report several moduli of smoothness and $K$-functionals, including recent results that give a fairly satisfactory characterization of best approximation by polynomials for functions in $L^p$ spaces, the space of continuous functions, and Sobolev spaces.