Fourier Analysis Second Semester 2008/9 Homework Assignment 3 (Due on 24th February 2009)

Only the questions marked with an asterisk (*) will count towards the assessment for this course. Most of these exercises are taken from Stein and Shakarchi.

1. Prove that l^{2}(Z) is complete, that is, every Cauchy sequence in l^{2}(Z) has a limit in l^{2}(Z).

2. This question follows on from question 3 on Assignment 2. Consider again the function f: [−π, π] → C defined by f(θ) = |θ|. Use Parseval’s Identity to prove

∑_{k=0}^{∞} \frac{1}{(2k + 1)^{4}} = \frac{π^{4}}{96} and ∑_{n=1}^{∞} \frac{1}{n^{4}} = \frac{π^{4}}{90} .

In fact, it is possible to calculate a formula for ∑_{n} 1/n^{k} for every even k. However, nobody has yet managed to do this for odd k, even for k = 3. If you have some spare time, perhaps you might want to try to evaluate ∑_{n} 1/n^{3}.

*3. Suppose that f is a continuously differentiable 2π-periodic function such that

∫_{−π}^{π} f(θ) dθ = 0.

(a) Use Parseval’s Identity to show that

∫_{−π}^{π} |f(θ)|^{2} dθ ≤ ∫_{−π}^{π} |f′(θ)|^{2} dθ.

(b) Find a function f_{0} which satisfies the same conditions as f but

∫_{−π}^{π} |f_{0}(θ)|^{2} dθ = ∫_{−π}^{π} |f′_{0}(θ)|^{2} dθ.

(c) Find a function f_{1} which satisfies the same conditions as f but

∫_{−π}^{π} |f_{1}(θ)|^{2} dθ < ∫_{−π}^{π} |f′_{1}(θ)|^{2} dθ.

(d) Suppose g is continuously differentiable and f is as above. Prove that

|∫_{−π}^{π} \overline{f(θ)}g(θ) dθ|^{2} ≤ ∫_{−π}^{π} |f(θ)|^{2} dθ ∫_{−π}^{π} |g′(θ)|^{2} dθ.

(e) Suppose h is continuously differentiable on [0, π] and h(0) = h(π) = 0. Prove

∫_{0}^{π} |h(θ)|^{2} dθ ≤ ∫_{0}^{π} |h′(θ)|^{2} dθ.

*4. The aim of this question is to evaluate the improper Riemann integral

∫_{0}^{∞} \frac{sin x}{x} dx = lim_{b→∞} ∫_{0}^{b} \frac{sin x}{x} dx.

(a) Show that

lim_{b→∞} ∫_{0}^{b} \frac{sin x}{x} dx = lim_{N→∞} ∫_{0}^{π} \frac{sin((N + 1/2)x)}{x} dx.

(b) Use the Riemann-Lebesgue Lemma to prove

lim_{N→∞} ∫_{−π}^{π} (\frac{1}{sin(x/2)} − \frac{2}{x}) sin((N + 1/2)x) dx = 0.

(c) Use the above and the fact that

∫_{−π}^{π} \frac{sin((N + 1/2)x)}{sin(x/2)} dx = 2π

for all N ≥ 1 to show that ∫_{0}^{∞} \frac{sin x}{x} dx = π/2.

5. Suppose f is 2π-periodic function which is k-times continuously differentiable. Prove that

f̂(n) = O(1/|n|^{k})

as |n| → ∞.

*6. Suppose f is 2π-periodic and integrable on [−π, π]. (a) Show that

f̂(n) = \frac{1}{4π} ∫_{−π}^{π}[f(x) − f(x + π/n)]e^{−inx} dx.

(b) Assume that f is Hölder continuous of order α, that is

|f(x) − f(y)| ≤ C|x − y|^{α}

for some C > 0 and all x and y. Use part (a) to prove

f̂(n) = O(1/|n|^{α}).