n
∑z
t=1
zn-1
z-----
z-1
(A.25)
and
V tz ∙ = z dLŸ ,'..!L .2,/1 - i cos(kπ /n ) 1
t! dz⅛ z-1 2 ^ sin(kπ /n) ,
(A.26)
Thus, taking the real part, we have
nn
(A.27)
COs cos(2kπ t/n) о 0, t cos(2kπ t/n) = 1 n,
t= 1 t-1 2
which proves the conditions (6) and (7). The other condition follow from the proof of Lemma
6 below. Q.E.D.
Proof of Lemma 4: We only prove (17); the other parts of Lemma 4 follow straightforwardly
from Lemmas 1-2. It is a standard exercise in linear algebra to verify that
(
Rq-d^ R
q r m q-r
RrTÂ R
rm
q-r
V 1
RqT, R
q r m r
RrTÂ R
r m r ^
л11
m
л21
m
1
<
Â22
m ^
(A.28)
where
22
A
m
n 2(n2RrTÂ R
\ r m r
(nRaTt R )(n2RT R )1(nRrT R )) 1
∖ q r m r-,q r m r' q r m q-r f
T ^ T 1 , T ^ ∖-1
- ( nRrTAmRq_r )(RqTÂmRq-r )^( nRqTAmRr))
(A.29)
= -n (RTÂ R )1( nRTÂ R )(л22/n2) = (Л21)T
q q m m q-r' q q r m r'q m 7 vm7
Therefore,
42