We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia .
 Berens, H.Lorentz, G. G.1972Inverse theorems for Bernstein polynomialsIndiana Univ. Math. J.21693–708.