an:06004438
Zbl 1238.41007
Holtz, Olga; Nazarov, Fedor; Peres, Yuval
New coins from old, smoothly
EN
Constr. Approx. 33, No. 3, 331-363 (2011).
0176-4276 1432-0940
2011
j
41A10 41A25
simulation; Bernstein-positive approximation; smoothness; Lorentz operators
Given a coin with unknown probability of heads \(p\), as well as a fair coin, the authors would like to simulate a coin with probability of heads \(f(p)\), where \(f:[0,1]\to (0,1)\) is a known function. First, the authors define the simulation rate for a simulation algorithm. Next, they recall some basic results regarding Bernstein polynomials, Bernstein basis, Bernstein coefficients, Bernstein-positive consistent approximation from below, Bernstein-positive consistent approximation from above. The relationship between Bernstein-positive approximation and smoothness is then established. Next, Lorentz operators and simultaneous approximation are examined. An iterative construction of Bernstein-positive consistent approximations schemes is very clean presented. Finally, the authors prove that \textit{G. G. Lorentz}'s Claim 10 [Math. Ann. 151, 239--251 (1963; Zbl 0116.04602)] is invalid.
Dan Bārbosu (Baia Mare)
0116.04602