Polynomial approximation with doubling weights

Abstract

A nonnegative function \({w \in \mathbb{L}_1[-1, 1]}\) is called a doubling weight if there is a constant L such that \({w(2I) \leqq L w(I)}\), for all intervals \({I \subset [-1, 1]}\), where 2I denotes the interval having the same center as I and twice as large as I, and \({w(I) := \int_I w(u) du}\). In this paper, we establish direct and inverse results for weighted approximation by algebraic polynomials in the \({\mathbb{L}_p, 0 < p \leqq \infty}\), (quasi)norm weighted by \({w_n := \rho_n {(x)}^{-1} \int_{x - \rho_n {(x)}}^{x + \rho_n {(x)}} w(u) du}\), where \({\rho_n {(x)} := n^{-1} \sqrt{1 - x^2} + n^{-2}}\) and w is a doubling weight.