Séries de Fourier 1. Définitions et notations. Premières propriétés

C(2π)CR2π
||f||∞= sup
R
|f|= sup
[0,2π]
|f| C
T f 2π ϕ(x) = f(T
2πx)
Lp
(2π)1≤p < +∞2πR
C|f|p(0,2π)||f||p=¡1
2πR2π
0|f|pdt¢1/p
(f|g) = 1
2πR2π
0f¯gdt L2
(2π)
n∈Zent7→ eint
(en)n∈ZL2
(2π)
P= Vect(en,n∈Z) = {P
finie
anen,an∈C}
f∈L1
(2π)n∈Zieme cn(f)(en)n
cn(f) = 1
2πZ2π
0
f(t)e−intdt = 1
2πZa+2π
a
f(t)e−intdt ,∀a∈R.
f∈L2
(2π)cn(f) = (f|en)
f∈L1
(2π)P∞
−∞ cn(f)eint f
SN=P+N
−Ncn(f)eint f x0
[a, b] (SN)Nx0
[a, b]
SN=a0
2+PN
n=0(ancos nx +bnsin nx)
an=an(f) = cn(f) + c−n(f) = 1
πR2π
0f(t) cos ntdt bn=i(cn(f) + c−n(f) = 1
πR2π
0f(t) sin ntdt
f n ∈Z7→ cn(f)bn(f) = 0 ,∀n≥1
an(f) = 0 ,∀n≥0
cn(f) = c−n(f)
f∈L2
(2π)cn(f) = (f|en)→0|n| → +∞
f∈L1
(2π)
f∈L1(a, b)−∞ ≤ a < b ≤ ∞ λ
lim
|λ|→+∞Zb
a
f(t)eiλtdt = lim
|λ|→+∞Zb
a
f(t) cos λt dt = lim
|λ|→+∞Zb
a
f(t) sin λt dt = 0
f∈L1
(2π)=⇒cn(f)→0|n| → +∞
f, g ∈L1
(2π)cn(f) = cn(g),∀n∈Zfpp
=g
(en)n∈ZL2
(2π)
f∈L2
(2π)P∞
−∞ cn(f)eint f L2
||f||2
2=P
n∈Z
|cn(f)|2
f∈L1
L1f
∃f2πsup
N
|SN(0)|= +∞

n≥0Dn:= Pn
k=−nek
n≥1Fn=D0+... +Dn−1
n
σn(f)σn(f) = S0+... +Sn−1
n
Dn1
2πRπ
−πDn(t)dt = 1 Dn(x) = sin((n+1
2)x)
sin(x/2) Sn(f) = f∗Dn, f ∈L1
2π
Fn1
2πRπ
−πFn(t)dt = 1 Fn(x) = 1
n¡sin(nx/2)
sin(x/2) ¢2σn(f) = f∗ Fn, f ∈L1
2πn≥1
f∈ C(2π)(σn(f))nfR
f∈Lp
(2π)1≤p < ∞(σn(f))nf||.||p
f∈ C(2π)a∈Rsi Sn(f)(a)→` ` =f(a)
f∈ C(2π)C1f=P∞
−∞ cn(f)en
Dn
f∈L1
(2π)a∈R
∃lim
t→0+f(a+t) := f+∃lim t→0+f(a−t) := f−
∃lim
t→0+
f(a+t)−f+
t∃lim
t→0+
f(a−t)−f+
t
(Sn(f)(a))n1
2(f++f−)