470 D. Positive Definite FunctionsProve that there exists a unique vector(, called the circumcenter of V, such
that V is contained in the closed ball with center ( and radius ro. Prove
also that the circumcenter ( is in the closed convex hull of V.
Exercise D.2. The complexification of a real Hilbert space ?-{fill is He =
1-{Yill + HHYilL with the obvious complex inner product. Observe that one
can complexify an orthogonal representation of a group on H"JR. Conversely,
the "realification" of a complex Hilbert space 1-{ is the real Hilbert space
1-{ with the real inner product ~( ·, · ). Observe that one can "realify" a
unitary representation of a group on H.
We outline below another proof of Schoenberg's Theorem.Exercise D.3. For a real Hilbert space 1-{ and arbitrary vector e E H, we
define
e®n
EXP(H) =CO EB~ H®n and EXP(e) = 0 + ~ Vnf'
where n is a fixed unit vector. Observe that (EXP(77),EXP(e)) = exp(11,e).
Check that if we define a unit vector
1
fr = exp(-2'y^2 llell^2 ) EXP(l'e) E EXP(H)
for/> 0 and e EH, then we have (77'Y, e'Y) = exp(-'y21177 - ell^2 /2).
Suppose that () is an action of r on 1-{ by affine isometries. We define a
representation u of r on JC,= span{EXP(e) : e EH} c EXP(H) by
~ ~ 1 1
u(s) L..,/\EXP(ei) = L..,,\iexp(2fleill^2 - 211e(s)eill^2 )EXP(()(s)ei)·
i i
Prove u is a well-defined orthogonal representation such that (u(s)O, 0) =
exp(-llb(s)ll^2 /2), where b(s) E 1-{ is the translation part of the affine isome-
try e(s).
References. Most of the results in this appendix are classical. Exceptions
are Proposition D.6, which comes from [24], and Lemmas D.7-D.9, which
come from [80]. Many other important results on Herz-Schur multipliers
are found in [47, 54, 78, 80]; see also [151].