Approximation of Sobolev classes by polynomials and ridge functions

Let W-p(r)(B-d) be the usual Sobolev class of functions on the unit ball B-d in R-d, W-p(o,r) (B-d) be the subclass of all radial functions in W-p(r)(B-d). We show that for the classes W-p(o,r)(B-d) and W-p(r)(B-d), the orders of best approximation by polynomials in L-q(B-d) coincide. We also obtain exact orders of best approximation in L-2(B-d) of the classes W-p(o,r)(B-d) by ridge functions and, as an immediate consequence, we obtain the same orders in L-2(B-d) for the usual Sobolev classes W-p(r)(B-d). (C) 2008 Elsevier Inc. All rights reserved.