Polynomial approximation inL p (0 |
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| Authors: | Ronald A. DeVore Dany Leviatan Xiang Ming Yu |
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| Affiliation: | 1. Department of Mathematics, University of South Carolina, 29208, Columbia, South Carolina, USA
2. Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel
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| Abstract: | We prove that forf∈L p , 0<p<1, andk a positive integer, there exists an algebraic polynomialP n of degree ≤n such that $$left| {f - P_n } right|_p leqslant Comega _k^varphi left( {f,frac{1}{n}} right)_p $$ whereω k ? (f,t)p is the Ditzian-Totik modulus of smoothness off inL p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP n can also be taken to be nondecreasing. |
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