HOMEWORK ASSIGNMENT 1: THE FOURIER TRANSFORM
Exercise 1. (S(Rn ) is closed under convolution)
Given f, g ∈ S(Rn ) show that f ∗ g ∈ S(Rn ):
a) Directly from the definition.
b) Using the Fourier transform.
R
Exercise 2. Let f ∈ L2 (Rn ) and let φ ∈ L1 (Rn ) with Rn φ(x) dx = 1 be given. We recall that,
given > 0, we define φ (x) := 1n φ( x ). Using only properties of the Fourier transform, show that:
f ∗ φ → f in L2 as → 0.
Exercise 3. (An explicit form of the Fourier transform on L2 )
Given f ∈ L2 (Rn ), show that the following limits hold in the L2 sense:
Z
(1)
fb(ξ) = lim
f (x) · e−2πix·ξ dx
M →∞
|x|≤M
and
Z
(2)
fb(ξ) · e2πix·ξ dξ.
f (x) = lim
M →∞
|ξ|≤M
[Hint: In order to prove (2), use the density of S(Rn ) in L2 (Rn ).]
We note that (1) gives us an explicit formula for the L2 Fourier transform (which doesn’t involve
finding an approximating sequence in S(Rn ) ). The identity in (2) is the analogue of the Fourier
inversion formula for the L2 Fourier transform.
Exercise 4. (The Fourier inversion formula on L1 )
Given f ∈ L1 (Rn ), we will show that:
Z
2
(3)
fb(ξ) · e2πix·ξ · e−π|ξ| dξ → f in L1 as → 0.
Rn
a) Explain why, given f ∈ L1 (Rn ) and > 0, the left-hand side of (3) makes sense.
R
R
b) Use the formula φb · ψ dx = φ · ψb dx to write the left-hand side of (3) as an integral involving f . Make sure to rigorously justify each step.
c) By writing the result in the appropriate way, deduce the claim in (3).
2
d) Can one replace e−π|ξ| in (3) by some other function (which is not a rescaled Gaussian) ?
Exercise 5. (The heat equation on Rn )
1
2
HOMEWORK ASSIGNMENT 1: THE FOURIER TRANSFORM
a) Given f ∈ S(Rn ), use the Fourier transform (in x) to solve the following initial value problem:
(
∂
n
∂t u − ∆u = 0 on Rx × (0, +∞)t
ut=0 = f on Rnx .
Write the solution as u(x, t) = Kt ∗ f (x), for some function Kt ∈ S(R). Determine the function Kt
explicitly.
b) Show that ku(x, t) − f (x)kL2 (Rnx ) → 0 as t → 0+.
Exercise 6. (The uncertainty principle revisited)
In this exercise, we give a quantitative version of the uncertainty principle. We recall that this
principle states that it is not possible for a function f and for its Fourier transform fb to be simultaneously localized in space and in frequency, respectively.
Let f ∈ S(R) be a function such that
R +∞
−∞
|f |2 dx = 1.
a) Show that:
Z
+∞
Z
x2 · |f (x)|2 dx ·
+∞
1
ξ 2 · |fb(ξ)|2 dξ ≥
.
16π 2
−∞
−∞
0
R +∞
[Hint: Use integration by parts to deduce that 1 = − −∞ x |f (x)|2 dx, from where it follows that
R +∞
1 ≤ 2 −∞ |x| · |f (x)| · |f 0 (x)| dx. Group terms appropriately in the Cauchy-Schwarz inequality.]
b) More generally, show that for all x0 , ξ0 ∈ R, it is the case that:
Z +∞
Z +∞
(x − x0 )2 · |f (x)|2 dx ·
(ξ − ξ0 )2 · |fb(ξ)|2 dξ ≥
−∞
−∞
1
.
16π 2
c) For which functions does equality hold in b)?
d) Give a one-sentence interpretation of b) as a version of the uncertainty principle.