Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation

For any x ε{lunate} (0, 1) we first prove that if f{hook}_x(t) ≡ |t - x| on [0, 1] then the Bernstein polynomials of f{hook}_x satisfy the asymptotic relation ∑_{k = 0}ⁿ | k n - x|(_kⁿ) x^k(1 - x) ^{n - k} = (2x (1 - x) π) ^{1 2} 1 √n + O( 1 n). This asymptotic relation is then used to study the rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. An estimate of the rate of convergence is given. This estimate is asymptotically the best possible at points where f{hook}′ is continuous.