Fourier Analysis Second Semester 2009/10 Homework Assignment 2 (Due on 9th February 2010, please staple muliple sheets together)

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. For δ ∈ (0, π), let f be defined on [−π, π] by

f(θ) = \{ \begin{matrix}0, & if |θ| > δ, \\ 1 − |θ|/δ, & if |θ| ≤ δ. \end{matrix}

(a) Plot the graph of f.

(b) Show that

f(θ) = \frac{δ}{2π} + 2 ∑_{n=1}^{∞} \frac{1 − cos(nδ)}{n^{2}πδ} cos(nθ).

2. Let f be the function given by f(θ) = |θ| for θ ∈ [−π, π].

(a) Draw the graph of f.

(b) Calculate the Fourier coefficients of f and show that

f̂(n) = \{ \begin{matrix}π/2, & if n = 0, \\ \{(−1)^{n} − 1\}/(πn^{2}), & if n ≠ 0. \end{matrix}

(c) What is the Fourier series of f in terms of sines and cosines?

(d) Taking θ = 0, prove that

∑_{k=0}^{∞} \frac{1}{(2k + 1)^{2}} = \frac{π^{2}}{8} and ∑_{n=1}^{∞} \frac{1}{n^{2}} = \frac{π^{2}}{6} .

3. Suppose {a_{n}}_{n=1}^{N} and {b_{n}}_{n=1}^{N} are two finite sequences of complex numbers. Let B_{k} = ∑_{n=1}^{k} b_{n} denote the partial sum of the series ∑_{n} b_{n} for k ≥ 1 and define B_{0} = 0.

(a) Summation by Parts. Prove the formula

∑_{n=M}^{N} a_{n}b_{n} = a_{N}B_{N} − a_{M}B_{M−1} − ∑_{n=M}^{N−1}(a_{n+1} − a_{n})B_{n}

(b) Deduce from the above formula the following lemma: Lemma (Dirichlet’s convergence test). If the partial sums of the series ∑_{n} b_{n} are bounded (that is, B_{k} ≤ C for all k ≥ 1) and {a_{n}}_{n} is a sequence of real numbers that decrease monotonically to 0, then ∑_{n} a_{n}b_{n} converges.

4. Consider the function f: [−π, π] → C defined by

f(x) = \{ \begin{matrix}−\frac{π}{2} − \frac{x}{2}, & if − π ≤ x < 0, \\ 0, & if x = 0, \\ \frac{π}{2} − \frac{x}{2}, & if 0 < x ≤ π. \end{matrix}

Draw the graph of f. Prove that the Fourier series of f is

\frac{1}{2i} ∑_{n≠0} \frac{e^{inx}}{n}

and prove that it converges pointwise to f, even though f is not continuous. [Hint: Use Dirichlet’s convergence test.]

5. Suppose that {f_{n}}_{n} is a sequence of integrable functions on [−π, π] such that

∫_{−π}^{π} |f_{k}(x) − f(x)| dx → 0

as k → ∞ for another integrable function f: [−π, π] → C. Show that f̂_{k}(n) → f̂(n) uniformly in n as k → ∞.

6. Let f be a continuous 2π-periodic function.

(a) Show that

f̂(n) = \frac{−1}{2π} ∫_{−π}^{π} f(x − (π/n))e^{−inx} dx.

(b) Prove that

lim_{|n|→∞} f̂(n) = 0.

[Hint: A continuous periodic function is uniformly continuous. You may use this fact without proof.]

(c) Using part (b), prove that if g is an integrable function on [−π, π], then

lim_{|n|→∞} ĝ(n) = 0.

This is called the Riemann-Lebesgue Lemma. *7. 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 (as stated in question 6) 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.

8. Suppose that f is an integrable 2π-periodic function. Prove that for each N

∫_{−π}^{π} S_{N}(f)(x) dx = ∫_{−π}^{π} f(x) dx.