UPMC Basic functional analysis
Master 1, MM05E 2011-2012
Fourier transform of integrable and square integrable functions
1) Letf ∈L1(R). The Fourier transform off is defined onRbyfˆ(ξ) =R+∞
−∞ f(x)e−2iπξxdx.
1. Letf(x) =e−πx2 for allx∈R. Compute dξdfˆ(ξ). Deducefˆ(ξ).
2. Letg(x) =e−|x|for allx∈R. Computeˆg.
3. Deduce the Fourier transform ofR3x7→h(x) = 1+x12.
2) a) Letχbe the characteristic function of the interval[−12,12]andφ=χ∗χ. Computeφ, φband χ.b b) Find two functions f andg∈L2(R), non identically zero and such thatf∗g= 0.
c) Show that there is no functionu∈L1(R)such thatf ∗u=f,∀f ∈L1.
3) LetA⊂Rn be a (Lebesgue) measurable set such that0<Ln(A)<∞, and letχA be its characteristic function. Show thatχbA∈L2(Rn)but χbA∈/L1(Rn).
4) We consider the differential equation
u0(x) +λxu(x) = 0∀x∈R,
whereλ >0.
a) Findusuch thatu(0) = 1.
b) Transform the previous equation to get an equation solved byu.ˆ c) Deduceu.ˆ
5) Compute the Fourier transform of the following functions : 1. f1(x) = (1− |x|)χ[−1,1](x).
2. f2(x) =xne−xχ[0,+∞)(x), n∈N. 3. f3(x) =sin(πx)πx .
4. f4(x) =sin(πx)
πx
2 .
5. f5(x) =(1+2iπx)1 n+1, n∈N.
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