Unboundedness of the Lipschitz constants of best polynomial approximation (Q1587648)

This paper is a proof of a single noteworthy theorem that confirms a conjecture of \textit{M. S. Henry} and \textit{J. A. Roulier} [J. Approximation Theory 22, 85-94 (1978; Zbl 0428.41003)]. For \(f\in C[-1,1]\), let \(q_n(f)\) be the best uniform approximation to \(f\) from \(P_n\) -- the algebraic polynomials of degree \(n\) or less. It is known that there is a finite number \(L_nf\) such that for all \(g\in C[-1,1]\); \[ \|q_n(f)-q_n(g)\|\leq (L_nf) \|f-g\|. \] It is also easy to show that if \(f\in P_n\) then for all \(g\in C[-1,1]\); \[ \|q_n(f)-q_n(g)\|\leq 2 \|f-g\|. \] This work shows that if there is a finite number \(B\) such that for all \(g\in C[-1,1]\); \[ \|q_n(f)-q_n(g)\|\leq B \|f-g\|, \] then \(f\) is a polynomial.