Iterated Bernstein polynomial approximations of degree n for continuous
function which also use the values of the function at i/n, i=0,1,...,n, are
proposed. The rate of convergence of the classic Bernstein polynomial
approximations is significantly improved by the iterated Bernstein polynomial
approximations without increasing the degree of the polynomials. The same idea
applies to the q-Bernstein polynomials and the Szasz-Mirakyan approximation.
The application to numerical integral approximations is also discussed.