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Inverse theorem for best polynomial approximation in $ L\sb p,\;0<p<1$


Authors: Z. Ditzian, D. Jiang and D. Leviatan
Journal: Proc. Amer. Math. Soc. 120 (1994), 151-155
MSC: Primary 41A25; Secondary 41A10, 41A17, 41A27
MathSciNet review: 1160297
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Abstract: A direct theorem for best polynomial approximation of a function in $ {L_p}[ - 1,1],\;0 < p < 1$, has recently been established. Here we present a matching inverse theorem. In particular, we obtain as a corollary the equivalence for $ 0 < \alpha < k$ between $ {E_n}{(f)_p} = O({n^{ - \alpha }})$ and $ \omega _\varphi ^k{(f,t)_p} = O({t^\alpha })$. The present result complements the known direct and inverse theorem for best polynomial approximation in $ {L_p}[ - 1,1],\;1 \leqslant p \leqslant \infty $. Analogous results for approximating periodic functions by trigonometric polynomials in $ {L_p}[ - \pi ,\pi ],0 < p \leqslant \infty $, are known.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1160297-7
Keywords: Inverse theorems, best polynomial approximation, $ {L_p}$ spaces, $ 0 < p < 1$
Article copyright: © Copyright 1994 American Mathematical Society