A note on Lototsky–Bernstein bases

Abstract

Abstract In 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 ) $O(n^{-2})$ .

Keywords

• Lototsky–Bernstein operators
• Recursive approximation
• Absolute functions
• Convergence rate