On the Randomized Bernstein Approximation Theorem and the Order of Approximation

TheWeierstrass Approximation Theorem provides an important result by approximating a given continuous function defined on a closed interval to a polynomial function. The Weierstrass Approximation theorem has great practical as well as theoretical utility. The Bernstein approximation theorem states that the Bernstein polynomial of order n is based on a sequence of independent Bernoulli distributed random variables playing the role of the polynomial function in Weierstrass Approximation theorem. Bernstein's proof was simple and elegant, based on some results from probability theory. The primary goal of this paper is to randomize the Bernstein approximation theorem with approximation orders when the order n in the Bernstein polynomial is replaced by nonnegative integer-valued random variables N-n,N- n >= 1 with E(N-n)->infinity when n -> 8. The obtained results are extensions and generalizations of the classical ones.