, Volume 1, Issue 1, pp 173-181

Some results of bernstein and jackson type for polynomial approximation inL p-spaces

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Abstract

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 functionu in the norms of the weightedL w p (−1, 1) spaces.p is any real number between 1 and ∞, andw(x) is either the Chebyshev or the Legendre weight. The estimates are given in terms of the degreeN of the polynomials and of the smoothness ofu. They include and generalize some theorems of Jackson. Some Bernstein-type inequalities are also given.