A pr 2 00 9 New coins from old , smoothly

Given a (known) function f : [0, 1] → (0, 1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (1994) implies that such a simulation scheme with IPp(N < ∞) = 1 exists iff f is continuous. Nacu and Peres (2005) proved that f is real analytic in an open set S ⊂ (0, 1) iff such a simulation scheme exists with IPp(N > n) decaying exponentially for p ∈ S. We prove that for α > 0 non-integer, f is in the space C[0, 1] if and only if a simulation scheme as above exists with IPp(N > n) ≤ C∆n(p),where ∆n(x) := max{ √ x(1 − x)/n, 1/n}. The key to the proof is a new result in approximation theory: Let H n be the cone of homogeneous bivariate polynomials of degree n with non-negative coefficients. We show that a function f : [0, 1] → (0, 1) is in C[0, 1] if and only if f has a series representation ∑ ∞ n=1 Fn(x, 1 − x) with Fn(x, y) ∈ H n and ∑ k>n Fk(x, 1 − x) ≤ C∆n(x) for all x ∈ [0, 1] and n ≥ 1. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some φn(x, y) ∈ H n satisfy |f(x) − φn(x, 1 − x)| ≤ C∆n(x) for all x ∈ [0, 1] and n ≥ 1, then f ∈ C[0, 1].