Negative theorems on generalized convex approximation

Eli Aaron Passow and John A. Roulier

Vol. 65 (1976), No. 2, 437–447

DOI: 10.2140/pjm.1976.65.437

Abstract

In this paper we show that there exist functions f ∈ C[−1,+1] with all (r + 1)-st order divided differences uniformly bounded away from zero for r fixed (f[x₀,x₁,⋯,x_r+1] ≧ δ > 0 for fixed δ and all sets x₀ < ⋯ < x_r+1 in [−1,+1]), for which infinitely many of the polynomials of best approximation to f do not have nonnnegative (r + 1)-st derivatives on [−1,+1].

Mathematical Subject Classification 2000

Primary: 41A50

Milestones

Received: 13 November 1975

Revised: 15 March 1976

Published: 1 August 1976