Math 295a Fall Term 2001
Assignment 3
1. Compute the Fourier transorm with respect to the variable x∈Rof
the following functions:
(i) uy(x) = e−y|x|, y > 0.
(ii) u(x) = e−|x|2
2.
(iii) u(x) = y2−x2
(y2+x2)2, y > 0 .
2. Show that Gdefined through G(x, y) = 1
π
y
x2+y2for x∈Rand y > 0
is harmonic, that is, 4G= 0, and conclude that
ug(x, y) := Z∞
−∞
G(x−˜x, y)g(˜x)d˜x , (x, y)∈R×(0,∞)
represents a solution of
(4u= 0 in R×(0,∞)
u=gon R× {0}
for g∈L1(R). What is limy→∞ ug(·, y)?
3. Let f∈ S (Rn) with supp(f)⊂B(0, R) for R > 0. Show that ˆ
fis
holomorphic in ξ∈Cnand satisfies
|ˆ
f(ξ+iη)| ≤ cN
1
(1 + |ξ|2)N/2eR|η|,(ξ, η )∈Rn.
Define S1={x∈R2| |x|= 1}and Tn:= (S1)nand assume u∈L1(Tn),
that is, assume that uis an integrable 2πperiodic function of all its vari-
ables, which can therefore be identified with a function defined on the n-
dimensional torus Tn. Then its Fourier transform is given by
F(u)(k) := ˆu(k) := 1
(2π)nZTn
e−ik·θu(θ)dθ .
In analogy to the Fourier transform introduced in class one has that
F ∈ LL1(Tn),l∞(Zn),
where l∞(Zn) is the Banach space of bounded sequences.
./.
