THE WEIERSTRASS APPROXIMATION THEOREM AND LARGE DEVIATIONS

Bernstein's proof (1912) of the Weierstrass approximation theorem, which states that the set of real polynomials over [0,1] is dense in the space of all continuous real functions on [0,1], is a classic application of probability theory to real analysis that finds its way into many textbooks ([1] and [2]) and journals [3]. All that is invoked in Bernstein's proof (at least as presented in [1] and [3]) is Chebyschev's inequality, and if the argument is applied to a function satisfying a Lipschitz condition, the rate of convergence of the Bernstein polynomials to the function can be shown to be at least of order l/nl/3. If instead of Chebyschev's inequality we use another probabilistic tool very much in vogue nowadays, the theoq7 of large deviations, we can prove that the rate of convergence is at least of order lnl/2 n/nl/2. All the material used here concerning large deviations is elementaq7 and can be found in [1]. Let f be a real function on [0,1] that satisfies a Lipschitz condition, i.e., there is a constant C such that for all x, y E [O, 1]