ƒign—ux dis™rets périodiques et tr—nsform—tion de pourier dis™rète

Z/nZ C
n > 1G
Z/nZ
C
GCF
Z R
R/Z
• F
•
G
•F

f
t f
t
•
•
•Z
f
•
F
FZ/nZ C
F
” + ” ” ×”
”.”
F
a∈G ea
ea(x) = 1si x =a,
0si x 6=a.
(ea)0≤a≤n−1F F
n f ∈ F
f=
n−1
X
a=0
f(a)ea.

Pn−1
a=0 λaea= 0
x∈G
n−1
X
a=0
λaea(x) = 0,
λx= 0
f
f f (ea)0≤a≤n−1
f(f(ea))0≤a≤n−1
F
h∈G ThF
Th(f)(x) = f(x+h).
Th= (T1)h,
χ T1
h∈G Thx∈G h ∈G
χ(x+h) = λhχ(x).
f(0) = 0
T1f
T1(f)(x) = λf(x),
f(x+ 1) = λf(x).

f(0) = 0 f(x) = 0 x
f= 0
χ χ(0) = 1
χ(0) = 1 x0
χ(h) = λh,
χ(x+h) = χ(x)χ(h).
G
GC∗χ G C∗
χ(x+y) = χ(x)χ(y).
G n χuu∈G
χu(v) = e2iπuv
n.
G1 + ··· + 1 n1
χ(1)n=χ(0) = 1. χ(1)
χ(1) χ(x)x
x1χ(x) = χ(1)x
C∗1
G
U1
F
ThThχu
e2iπhu
n.

HFt
H(Tt(f)) = Tt(H(f)),
H◦Tt=Tt◦H.
H
χ λ T1χ
T1(H(χ)) = H(T1(χ)) = H(λχ) = λH(χ).
H(χ) = 0 H(χ)
T1λ H(χ) = µχ
H
µ=H(χ)(0)
H T1
H=
n−1
X
i=0
aiTi
1.
T
n T H
T H
H=
n−1
X
i=0
aiTi.