On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
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SIZES OF COMPACT SUBSETS OF HILBERT SPACE 317<br />
Now (b) is equivalent by <strong>the</strong> zero-one law ([13] A, p. 228) to <strong>the</strong><br />
following: for any E > 0,<br />
or<br />
i L<br />
n-1 I<br />
eox2J2 dx < co.<br />
00<br />
e-x2/2 dx is asymptotic to e-M2/2/M.<br />
As is well known, an integration by parts shows that as M --+ co,<br />
s M<br />
Thus (b) is equivalent to<br />
5 b, exp (- c2/2bn2) < co.<br />
n-1<br />
Letting b, = olyr (log n)-l12, n > 2, we obtain <strong>the</strong> series<br />
f c$(log 4-l/2 d’~,a. (63)<br />
n=2<br />
If (c) holds, i.e., ollt --t 0, <strong>the</strong>n <strong>the</strong> terms <strong>of</strong> (6.8) become less than<br />
W-~ for n large, so (b) holds.<br />
Conversely suppose (c) is false, so that for some 6 > 0, ol, > 6<br />
for arbitrarily large values <strong>of</strong> 7t. For such an n <strong>and</strong> nil2 42.<br />
>, & - ?.w - 1)/2?21’2 log n + co<br />
as n + co (recall that 6 is independent <strong>of</strong> n). Thus (6.8) diverges <strong>and</strong><br />
(b) fails, so (b) 3 (c). Q.E.D.