Fourier Analysis Second Semester 2008/9 Homework Assignment 2 (Due on 6th 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. Recall the Dirichlet kernel:

D_{N}(x) = ∑_{n=−N}^{N} e^{inx} = \frac{sin((N + \frac{1}{2})x)}{sin(x/2)}.

Show that D_{N} is an even function and that

∫_{−π}^{π} D_{N}(x) dx = 1.

2. 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).

*3. 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} .

*4. 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.

*5. 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.]

6. 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 → ∞.