Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation
Let \(S_ n(x)\) be the Bernstein polynomial of degree n for the function \(f_ x(t):=| t-x|\) evaluated at \(t=x\). The authors first show that \(S_ n(x)=(2x(1-x)/\pi n)^{1/2}+O(1/n)\) and use this to give the estimate \(| B_ n(f,x)-f(x)-\sigma (x(1-x)/2\pi n)^{1/2}| =O(1/n)\) for the approximation of functions f with derivative \(f'\) of bounded variation by Bernstein polynomials \(B_ n(f,x)\) of degree n. Here \(\sigma\) is the jump \(\sigma =f'(x+0)-f'(x-0)\) in the derivative at the position x. The authors also give expressions for the O(1/n) on the right-hand side in terms of the total variation of certain functions associated with \(f'\).
scientific article; zbMATH DE number 4127801