Error bounds for Romberg quadrature

Denote by R [f] the error of a Romberg quadrature rule applied to the funct ion f. We determine approximately the constants in the bounds of the types \ R [f]\ I const sup \ f((mu)) (x)\ and R [f] less than or equal to constVa r f((mu-1)) (mu = 1, 2,...) for all classical Romberg rules. By a compariso n with the corresponding constants of the Gaussian rule we give the stateme nt "The Gaussian quadrature rule is better than the Romberg method" a preci se meaning.