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Polynomial approximation inL p (0<p<1)

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Constructive Approximation Aims and scope

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

$$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$

whereω ϕ 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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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, 29208, Columbia, South Carolina, USA

    Ronald A. DeVore & Xiang Ming Yu

  2. Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Dany Leviatan

Authors
  1. Ronald A. DeVore
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  2. Dany Leviatan
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  3. Xiang Ming Yu
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Communicated by Vilmos Totik.

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DeVore, R.A., Leviatan, D. & Yu, X.M. Polynomial approximation inL p (0<p<1). Constr. Approx 8, 187–201 (1992). https://doi.org/10.1007/BF01238268

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  • DOI: https://doi.org/10.1007/BF01238268

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