Math 295a Fall Term 2001

Assignment 3

1. Compute the Fourier transorm with respect to the variable x ∈ ℝ of the following functions:

(i) u_{y}(x) = e^{−y|x|} , y > 0.

(ii) u(x) = e^{− \frac{|x|^{2}}{2}} . (iii) u(x) = \frac{y^{2}−x^{2}}{(y^{2}+x^{2})^{2}} , y > 0 .

2. Show that G defined through G(x, y) = \frac{1}{π} \frac{y}{x^{2}+y^{2}} for x ∈ ℝ and y > 0 is harmonic, that is, △G = 0, and conclude that

u_{g}(x, y) := ∫_{−∞}^{∞} G(x − x̃, y)g(x̃) dx̃ , (x, y) ∈ ℝ × (0, ∞)

represents a solution of

\begin{cases}△u & = 0 in ℝ × (0, ∞) \\ u & = g on ℝ × \{0\} \end{cases}

for g ∈ L_{1}(ℝ). What is lim_{y→∞} u_{g}(·, y)?

3. Let f ∈ S(ℝ^{n}) with supp(f) ⊂ 𝔹(0, R) for R > 0. Show that f̂ is holomorphic in ξ ∈ ℂ^{n} and satisfies

|f̂(ξ + iη)| ≤ c_{N} \frac{1}{(1 + |ξ|^{2})^{N/2}} e^{R|η|} , (ξ, η) ∈ ℝ^{n} .

Define 𝕊^{1} = {x ∈ ℝ^{2} | |x| = 1} and 𝕋^{n} := (𝕊^{1})^{n} and assume u ∈ L_{1}(𝕋^{n}), that is, assume that u is an integrable 2π periodic function of all its variables, which can therefore be identified with a function defined on the ndimensional torus 𝕋^{n}. Then its Fourier transform is given by

(F(u))(k) := û(k) := \frac{1}{(2π)^{n}} ∫_{𝕋^{n}} e^{−ik·θ}u(θ) dθ .

In analogy to the Fourier transform introduced in class one has that

F ∈ L(L_{1}(𝕋^{n}), l_{∞}(ℤ^{n})) ,

where l_{∞}(ℤ^{n}) is the Banach space of bounded sequences.

./.

4. Let n = 1 and prove that

(S_{n}u)(θ) := ∑_{k=−n}^{n} û(k)e^{ik·θ} = \frac{1}{2π} ∫_{−π}^{π} u(θ − φ)D_{n}(φ) dφ

for D_{n}(θ) = ^{\frac{sin((n+1/2)θ)}{sin( \frac{θ}{2} )}}.

5. Show that

(S_{n}u)(0) → 0 (n → ∞)

if g ∈ L_{1}(−π, π) for g(θ) = f(θ)/θ and deduce that

(S_{n}u)(θ_{0}) → u(θ_{0}) (n → ∞)

if u is Lipscitz continuous at θ = θ_{0} .

The Homework is due by November 2 2001