Improved error bounds for near-minimax approximations

Abstract

Givenf εC (n+1)[−1, 1], a polynomialp n, of degree ≤n, is said to be near-minimax if $$\left\| {f - p_n } \right\|_\infty = 2^{ - n} |f^{(n + 1)} (\xi )|/(n + 1)!,$$ ((*)) for some ζ ε (−1,1). For three sets of near-minimax approximations, by considering the form of the error ∥f −p n∥∞ in terms of divided differences, it is shown that better upper and lower bounds can be found than those given by (*).