Weierstrass type approximation by weighted polynomials in Rd

Abstract 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 2 n with varying weights ( 1 + t 2 ) − 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 w α ( x ) ≔ ( 1 + | x 1 | α + . . . + | x d | α ) 1 α , 0 α 1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞ . Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold.