Approximation of Sobolev classes by polynomials and ridge functions Original Research Article

Let View the MathML sourceWpr(Bd) be the usual Sobolev class of functions on the unit ball BdBd in RdRd, and View the MathML sourceWp∘,r(Bd) be the subclass of all radial functions in View the MathML sourceWpr(Bd). We show that for the classes View the MathML sourceWp∘,r(Bd) and View the MathML sourceWpr(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 View the MathML sourceWp∘,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 View the MathML sourceWpr(Bd).