Some numerical reslts on best uniform polynomial approximation of. chi. sup. alpha. on (0,1)

Let {alpha} be a positive number, and let E{sub n}(chi{sup {alpha}}; (0,1)) denote the error of best uniform approximation to {chi}{sup {alpha}}, by polynomials of degree at most n, on the interval (0,1). The Russian mathematician S.N. Bernstein established the existence of a nonnegative constant {Beta}({alpha}) such that {Beta}({alpha}):= {sub n{yields}{infinity}lim(2n){sup 2{alpha}}E{sub n}({chi}{sup {alpha}};(0.1)). In addition, Bernstein showed that {Beta}{alpha} < {Gamma}(2{alpha}){vert bar}sin(pi}{alpha}){vert bar}/{pi} ({alpha} > 0) and that {Gamma}(2{alpha}){vert bar}sin({pi}{alpha}){vert bar}/{pi} (1{minus}1/2{alpha}{minus}1) < {Beta}({alpha}) ({alpha} > {1/2}), so that the asymptotic behavior of {Beta}({alpha}) is known when {alpha}{yields}{infinity}. Still, the problem of trying to determine {Beta}({alpha}) more precisely, for all {alpha} > 0, is intriguing. To this end, we have rigorously determined the numbers for thirteen values of {alpha}, where these numbers were calculated with a precision of at least 200 significant digits. For each of these thirteen values of {alpha}, Richardson's extrapolation was applied to the products to obtain estimates of {Beta}({alpha}) to approximately 40 decimal places. Included are graphs of the points ({alpha},{Beta}({alpha})) for the thirteen values of {alpha} that we considered