A note on Lototsky–Bernstein bases

Abstract

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