ON THE DENSITY OF MULTIVARIATE POLYNOMIALS

In this paper we consider multivariate approximation by weighted polynomials of the form w gamma n (X)pn(X), where pn is a multivariate polynomial of degree at most n, w is a given nonnegative weight with nonempty zero set, and-yn up arrow infinity. We study the question if every continuous function vanishing on the zero set of w is a uniform limit of weighted polynomials w gamma n (X)pn(X). It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that-yn = o(n).