Preface

If f 2 CŒ0; 1 the convergence is uniform. This result shows that the rate of convergence Bn.f ; x/ f .x/ ! 0 as n ! 1 is of order not better than 1=n if f 00.x/ ¤ 0: In this sense the theorem of E. Voronovskaja is a first example of saturation, i.e. the optimal rate of convergence is 1=n: To increase the rate of convergence, it was P. L. Butzer [34] who introduced in 1953 the following linear combinations: