
79Superefcient Monte Carlo Simulations
Example 3.3
2D Walsh system. The Walsh system is a complete orthonormal set associated with the doubling
map (3.10). The Rademacher system on the unit interval equipped with Lebesgue measure is a set
of orthonormal functions
xr xx n{(x) sgn(sin2 )0,1 ,},
n
)
[
=π
+
(3.35)
where sgn is the signum function. Note that the Rademacher system is not complete. The com-
plete orthonormal basis is the given by the Walsh–Paley system
n 0
)
(3.36)
where W
0
(x) = 1 and
=
νν ν
Wx rxrx rx() () () ()
n
m12
, where n = 2
ν
1
+ 2
ν
2
+ ⋯ + 2
ν
m
is the binary
representation of n and ν
1
< ν
2
< ⋯ ν
m
. To show that ...