Interpolatory estimates for convex piecewise polynomial approximation

2018/11/02 by Kopotun, Kirill A., Leviatan, Dany, Shevchuk, Igor A.
#41A10 #41A25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.01087

Abstract

In this paper, among other things, we show that, given r∈ N, there is a constant c=c(r) such that if f∈ C^r[-1,1] is convex, then there is a number \mathcal N=\mathcal N(f,r), depending on f and r, such that for n≥\mathcal N, there are convex piecewise polynomials S of order r+2 with knots at the Chebyshev partition, satisfying |f(x)-S(x)|≤ c(r)( min\ 1-x², n^-1√(1-x²) \ )^r ω₂(f^(r), n^-1√(1-x²) ), for all x∈ [-1,1]. Moreover, \mathcal N cannot be made independent of f.

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