Dolomites Research Notes on Approximation

In the present paper we consider the following central problem on the approximation by homogeneous polynomials: For which 0-symmetric star like domains K ⊂ R d and which f ∈ C(∂ K) there exist homogeneous polynomials hn , hn+1 of degree n and n + 1, respectively, so that uniformly on ∂ K f = limn→∞ (hn + hn+1 )? This question is the analogue of the Weierstrass approximation problem when polynomials of total degree are replaced by the homogeneous polynomials. A survey of various recent results on the above question is given with some relevant open problems being included, as well.