Approximation of Sobolev classes by polynomials and ridge functions

Let Wpr(Bd) be the usual Sobolev class of functions on the unit ball BdBd in RdRd, and Wp∘,r(Bd) be the subclass of all radial functions in Wpr(Bd). We show that for the classes Wp∘,r(Bd) and Wpr(Bd), the orders of best approximation by polynomials in Lq(Bd)Lq(Bd) coincide. We also obtain exact orders of best approximation in L2(Bd)L2(Bd) of the classes Wp∘,r(Bd) by ridge functions and, as an immediate consequence, we obtain the same orders in L2(Bd)L2(Bd) for the usual Sobolev classes Wpr(Bd).