Some results of bernstein and jackson type for polynomial approximation in Lp-spaces

This work is motivated by the analysis of stability and convergence of spectral methods using Chebyshev and Legendre polynomials. We investigate the properties of the polynomial approximation to a function u in the norms of the weighted Lwp (-1, 1) spaces. p is any real number between 1 and ∞, and w(x) is either the Chebyshev or the Legendre weight. The estimates are given in terms of the degree N of the polynomials and of the smoothness of u. They include and generalize some theorems of Jackson. Some Bernstein-type inequalities are also given. © 1984 JJAM Publishing Committee.