1 Théorie L2 des séries de Fourier

C1
(Sn(f))
1pfp
2
2( ) 2( )
f f
2
E
h. , .i
x7→ ||x|| =phx , xi
E

E=2(X, T, µ) (X, T, µ)
∀(f, g)∈2( )2,hf , gi=ZX f¯g dµ.
2( ) E= ( ,B( ), λ )`2( ) E= ( ,P( ) , µ)µ
H M ⊂H
M H
M⊥={x∈H, ∀y∈M, hx , yi= 0}.
M⊥H
M
H=M⊕M⊥.
M H
M H →M
πM
M H
∀x∈H, ||πM(x)|| 6||x||
x∈M πM
M6={0}

x∈H πM(x)M
x M y M
||y−x|| = (x, M) = inf
z∈M||z−x||.
I I H
(ei)∈IH I
∀i, j ∈I2,hei, eji=δi,j .
(ei)i∈IH I M = Vect(ei)i∈I
∀x∈H, πM(x) = X
i∈Ihx , eiiei.
(ei)i∈IH I
MVect(ei)i∈IH
∀x∈H, πM(x) = X
i∈Ihx , eiiei.
P
i∈Ihx , eiiei.
(xi)i∈IH
X
i∈I||xi||2<∞.
I(xi)i∈IH
(In)n∈I
[
n∈
In=I.
X
i∈In
xi!n

H(In)
X
i∈I
xi.
(ei)i∈IH M = Vect(ei)i∈I
(αi)i∈I`2(I) (αiei)i∈I
(αi)i∈I(βi)i∈I`2(I)
*X
i∈I
αiei,X
i∈I
βiei+=X
i∈I
αi¯ βi
X
i
αiei
2
=X
i∈I|αi|2.
x∈H(hx , eii)`2(I)
πM(x) = X
i∈Ihx , eiiei.
x7→ (hx , eii)i∈I
M
H `2(I)
(ei)i∈I
Vect(ei)i∈I=H
∀x∈H, ∀i∈I, hx , eii= 0 ⇒x= 0.
H(ei)i∈I
H=M

H=2( )
f2( )
∀n∈,hf , eni=b f(n).
∀n∈,hf , eni=Zf(t)en(t)dt =Zf(t)e−n(t)dt =b f(n).
(en)n∈H
∀n, m ∈,hen, emi=δn,m
f2( )
∀n∈,hf , eni= 0.
∀n∈,b f(n) = 0.
f(en)
(en)
n>0f∈2( ) (Sn(f))
n f
2Tn
(ek)−n6k6nf2( )
Sn(f) = πTn(f)