On the Characters and the Plancherel Formula of Nilpotent Groups ...

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 325
Let 6, = Y(1/2”), defined and positive for n large enough (unless
y 3 0, in which case the conclusion is trivial). Then 6, 4 0 as n --+ co.
Now
so
NC, 1/w < &$znk,
H(C, l/2”) < log A + k log (l/S,).
Let .v~ = (log (1/S,J)li2. By Th eorem 3.1 it suffices to prove that
(Note how the dimension becomes irrelevant.)
Now
so
I
p)(e-2’) > l/2” for %a-1 < x < XT& ,
IN cp(e-““) a% >, g (xn+l - x,)/2”+’ =
n-N
*sg+l %P - f %n/2*+1
m--N
= 8 n ;+l %P - x,PN+‘,
so the required series converges. Q.E.D.
Fernique [7j shows that Theorem 7.1 is optimal of its kind in a
sense, even for k = 1, since if
i
m
fp(e-““) dx = + cc
and CJZ
satisfies some additional mild monotonicity assumptions, then
counterexamples to sample continuity exist. However, note that we
may take a process xf on T = [0, l] satisfying the hypotheses of
Theorem 7.1 and transform it by a “steep” homeomorphism f of T,
e.g.f(t) = l/log (l/t), into a process x!(t) which may no longer satisfy
7.1. (c) but of course is still sample-continuous. The E-entropy of the
range is unchanged, so Theorem 3.1 applies to xtfl) and has a broader
range of applications. Note however that such a transformation
destroys stationarity of the process, and for stationary processes
Theorem 7.1 may be essentially the best possible.