New estimates related with the best polynomial approximation

Abstract

In some old results, we find estimates the best approximation \(E_{n,p}(f)\) of a periodic function satisfying \(f^{(r)}\in\mathbb{L}^p_{2\pi}\) in terms of the norm of \(f^{(r)}\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \(f^{(r)}\in \mathbb{L}^q_{2\pi}\), with \(1<q<p<\infty\). We will present inequalities of the form \(E_{n,p}(f)\leq C(n)\Vert D^{(r)}f\Vert_q\), where \(D^{(r)}\) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.