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Some results of Bernstein and Jackson type for polynomial approximation in \(L^ p\)-spaces. (English) Zbl 0568.41006

Some extensions of Jackson’s and Bernstein’s theorems for polynomial approximations are presented. They concern both the best approximation error and the truncation error using Fourier series. The framework in which these results are given is that of the weighted \(L^ p\)-spaces, where p is any real between 1 and \(\infty\), and the weight function is either \(w(x)=(1-x^ 2)^{-1/2}\) (Chebyshev weight) or w(x)\(\equiv 1\) (Legendre weight). Some Bernstein type inequalities in the norms of the above spaces are also given. These results can be applied to the analysis of spectral methods for the numerical approximation of partial differential equations.

MSC:

41A10 Approximation by polynomials
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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