pete 
pai 
f and g are periodic with period À = M. 
As the scalar product is invariant under orthogonal transformations F, 
we can now approximate (6) with the equation 
(9/60) f.q(u) —/(0/80) £.6(0) - O; T — Ff? dq e Fg (3) 
The definition of gq(u) is cyclic in that g(u) is obtained from g(O) by 
moving all components upwards yu steps and filling in at the bottom 
with what falls out at the top. The difference between the relations 
(6) and (9) introduced through this definition is the wrap-around 
error. The number p indicates to what extent the components of g have 
been cyclically displaced. Introduce the cyclic linear unit displace- 
ment operator U: 
0°L0 07°, 10 
00:13 051 "0 
U = oH, UID BY, (10) 
0000 1 
1000. 0 
Direct calculation gives Ug(0) = g(1) and generally U"q(0) = q(u). 
Denoting g(0) with g, (9) can now be rewritten as 
(8/80) £.U"g = (3/8) F.D'& = 0 (11) 
where D - FUF '. In the special case when the transformation matrix F 
is chosen such as to make D diagonal, standard matrix theory gives 
F = (e, &5 «no ey 
D. = diag(A, Ay cod Ay) 
H 225 HB u 
D' = diag(X, As IY Ay) (12) 
- 641 -