A note on Lototsky–Bernstein bases.

In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | on [ − 1 , 1 ] by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n (x) to | x | preserves good shapes on [ − 1 , 1 ] . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O (n − 2) . [ABSTRACT FROM AUTHOR]