BEST UNIFORM RATIONAL APPROXIMATION OF VERTICAL-BAR-X-VERTICAL-BAR ON [-1, 1]

We consider best rational approximants in the uniform norm to the function absolute value of x on [-1, 1] . The main result is a proof of a conjecture by R. S. Varga, A. Ruttan, and A. J. Carpenter. They have conjectured that if E(nn) (absolute value of x, [-1, 1]) , n is-an-element-of N , denotes the error of the nth degree rational approximant, then lim(n-->infinity) e(pi square-root n) E(nn) (absolute value of x, [-1, 1]) = 8. This conjecture generalizes earlier results, among them most prominently results by D. J. Newman and by N. S. Vyacheslavov.