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 ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
SIZES OF COMPACT SUBSETS OF HILBERT SPACE 327<br />
Pro<strong>of</strong>. We can restrict ourselves to 0 < t < 2~. For any s <strong>and</strong> t<br />
in (0, 271),<br />
Let<br />
Jws - 4”) = E 1 rnfl /%A( cos 11s - cos nt) + fimTn(sin tls - sin &)I21<br />
= 2 f pm2(1 - cos (n(s - t))).<br />
n-1<br />
B” = ;l Pm2 = gl ti2, b = gl ti .<br />
Given E > 0, we choose a minimal M(E) such that<br />
For all x, 1 - cos x < x2, so if<br />
<strong>the</strong>n<br />
Hence<br />
1 s - t ) < 42@M,<br />
2 5 /3%2(1 - cos (n(s - t))) < q4<br />
n-1<br />
N(S, E) < 2 %h,BM/e + 1.<br />
Now M(C) < 2i for <strong>the</strong> least i such that<br />
For any 6 > 0,<br />
for l small enough by (5.2). Thus<br />
n({ti}, l 2/8b) < ( 1/e2)l+*<br />
M(r) < 2.~-~‘~+5’<br />
for E small enough. Hence r(S) < 2. Q.E.D.<br />
PROPOSITION 7.3. If C /Ia < co, <strong>the</strong>n series (1”) conwerges uni-
Change language