Weierstrass type approximation by weighted polynomials in Rd

In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ±∞ is a uniform limit on R of weighted algebraic polynomials of degree 2n with varying weights (1+t2)−n. We will verify a similar statement in the multivariate setting for a general class of convex weights. We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form wα(x)≔(1+|x1|α+…+|xd|α)[Formula presented],

Homogeneous polynomial approximation on convex and star like domains

In the present paper we consider the following central problem on the approximation by homogeneous polynomials: For which 0-symmetric star like domains K ⊂ ℝd and which f ∈ C(∂ K) there exist homogeneous polynomials hn, hn+1 of degree n and n + 1, respectively, so that uniformly on ∂ K (formula presented) 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. © 2023, Padova University Press. All rights reserved