Fourier series

UPMC Basic functional analysis

Master 1, MM05E 2011-2012

Fourier series

1) Letf :R→Cbe aT-periodic function such thatf ∈L²(0, T).

a) Show that

fˆ(k) := 1 T

Z T

0

f(t)e^−i2πkt/Tdt= 1 T

Z a+T

a

f(t)e^−i2πkt/Tdt ∀a∈R, ∀k∈Z.

b) Show that

f(t) =X

k∈Z

fˆ(k)e^i2πkt/T for a.e.t∈R.

c) Show that

1 T

Z T

0

|f(t)|²dt=X

k∈Z

|fˆ(k)|².

2) Letf be aT-periodic function such thatf ∈L²(0, T). Let

fˆT(k) := 1 T

Z T

0

f(t)e^−i2πkt/Tdt and fˆ2T(k) := 1 2T

Z 2T

0

f(t)e^−iπkt/Tdt

for allk∈Z. Show thatfˆ_2T(2k) = ˆf_T(k), andfˆ_2T(2k+ 1) = 0for allk∈Z.

3) Letgbe the2-periodic function whose restriction to[0,2)isg(x) = (1−x)χ_[0,1](x).

a) Expandg in Fourier series.

b) Find a2-periodic function such thatf⁰+f =g in(0,2).

4) Let us define the functionsPm:R→Rfor allm∈N^∗ in the following way :

• P₁ is the2π-periodic function such thatP₁(x) =π−xifx∈]0,2π].

• ∀m≥1, we defineP_m+1 in terms ofP_mby

P_m+1(x) =C_m+1+ Z x

0

P_m(t)dt,

where the constantC_m+1 is such that the average of P_m+1 over(0,2π)is zero.

a) Show that P_mis2π-periodic and thatP_m∈L²(0,2π)for allm∈N^∗. b) ExpandP_min Fourier series.

c) Deduce that

X

n∈N^∗

1 n² =π²

6 , X

n∈N^∗

1 n⁴ =π⁴

90.

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5) LetCper([0,2π]) be the space of all continuous functions from Rto R, which are 2π-periodic. We want to show that for allx∈[0,2π), the space of all functionsf ∈ C_per([0,2π])whose Fourier series diverges atx, defines aGδ dense subspace.

1. Prove the following refinement of the Banach-Steinhaus Theorem : Let(Ti)_i∈I be an arbitrary family of continuous linear maps from a Banach spaceEto a normed linear spaceF. Then we have the following alternative :

(a) sup_i∈I kT_ik_L(E,F)<+∞,

(b) For ally in aGδ dense subset ofE, sup_i∈I kTi(y)kF = +∞.

Let us fixx∈[0,2π). Forf ∈ Cper([0,2π]), we introduce the Fourier coefficientscn(f), and the partial Fourier sum :

SN(f)(x) := X

−N6n6N

cne^inx= 1 2π

Z 2π

0

f(t)DN(x−t)dt,

with

D_N(x) := X

−N6n6N

e^inx.

2. Show thatkDNk_L1(0,2π)→+∞whenN→+∞.

3. Show that

kS_N(·)(x)k_{L(Cper([0,2π]),R)}= 1

2πkD_Nk_L1(0,2π).

4. Conclude. Show that on aGδ dense subspace of Cper([0,2π])the Fourier series diverges.

5. Show that the mapf ∈L¹(0,2π)→(cn(f))n∈Z∈c0, wherec0 is the space of all sequences converging to zero, is not surjective.

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