ArticleOriginal scientific textA counterexample in comonotone approximation in
Title
A counterexample in comonotone approximation in space
Authors 1, 2
Affiliations
- Department of Mathematics, Hangzhou University, Hangzhou, Zhejiang, P.R. China
- Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G.2G1
Abstract
Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function , with for x ∈ [0,1] and for x ∈ [-1,0], such that lim sup_{n→∞} (e_n^{(k)}(f)_p) / (ω_{k+2+[1/p]}(f,n^{-1})_{p}) = + ∞ where is the best approximation of degree n to f in by polynomials which are comonotone with f, that is, polynomials P so that for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution to the converse result in comonotone approximation in space for 1 < p ≤ ∞.
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