Fourier Analysis Second Semester 2009/10 Homework Assignment 1 (Due on 26th January 2009)

Questions marked with an asterisk will form part of the assessment for this course. 1. For z ∈ C, we define the complex exponential by

e^{z} = ∑_{n=0}^{∞} \frac{z^{n}}{n!}

(a) Show that the above definition makes sense by showing that the series converges for every complex number z.

(b) If z_{1} and z_{2} are two complex numbers, prove that e^{z_{1}}e^{z_{2}} = e^{z_{1}+z_{2}}. [Hint: You may like to use the binomial theorem.]

(c) Show that

e^{iy} = cos y + i sin y

whenever y ∈ R. This is called Euler’s identity. [Hint: Take sine and cosine to also be defined via their power series.]

(d) More generally, prove that

e^{z} = e^{x}(cos y + i sin y),

where z = x + iy and x, y ∈ R.

2. Verify that the function x ↦ e^{inx} is periodic with period 2π and that

\frac{1}{2π} ∫_{−π}^{π} e^{inx} dx = \{ \begin{matrix}1, & if n = 0, \\ 0, & if n ≠ 0. \end{matrix}

Use this fact to prove that if n, m ≥ 1 we have

\frac{1}{π} ∫_{−π}^{π} cos nx cos mx dx = \{ \begin{matrix}1, & if n = m, \\ 0, & if n ≠ m. \end{matrix}

and similarly

\frac{1}{π} ∫_{−π}^{π} sin nx sin mx dx = \{ \begin{matrix}1, & if n = m, \\ 0, & if n ≠ m. \end{matrix}

Finally, show that

\frac{1}{π} ∫_{−π}^{π} cos nx sin mx dx = 0

for any n and m. [Hint: Calculate e^{inx}e^{−imx} + e^{inx}e^{imx} and e^{inx}e^{−imx} − e^{inx}e^{imx}.] *3. In lectures we arrived at the differential equation

y′′(t) + c^{2}y(t) = 0  (†)

as a model for the displacement y(t) of a mass attached to a spring as time t passes.

(a) Show that the function

y(t) = a cos(ct) + b sin(ct)  (‡)

solves (†), where a, b ∈ C.

(b) Show that any solution y of (†) which is twice continuously differentiable is of the form (‡). [Hint: Start by differentiating the two functions g(t) = y(t) cos(ct)−c^{−1}y′(t) sin(ct) and h(t) = y(t) sin(ct) + c^{−1}y′(t) cos(ct).]

4. Suppose F is a function on (a, b) with two continuous derivatives. Show that whenever x and x + h belong to (a, b), one may write

F (x + h) = F (x) + hF′(x) + \frac{h^{2}}{2} F′′(x) + h^{2}φ_{x}(h)

where φ_{x}(h) → 0 as h → 0 for each x. Deduce that, for each x,

\frac{F (x + h) + F (x − h) − 2F (x)}{h^{2}} → F′′(x)

as h → 0.

*5. In lectures we have defined the Fourier sine cofficients of a function f as

A_{n} = \frac{2}{π} ∫_{0}^{π} f(x) sin(nx) dx

for n = 1, 2, 3, . . .. Show that

A_{n} = \frac{2h}{n^{2}} \frac{sin(np)}{p(π − p)}

when f: [0, π] → C is defined as

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

Observe, this function f may be interpreted as the initial position of a plucked string, and so is a relevant example given our physical motivation.

*6. This excercise will show how the symmetries of a function imply certain properties of Fourier coefficients. Let f be a 2π-periodic integrable function defined on R.

(a) Show that the Fourier series of f can be written as

f̂(0) + ∑_{n=1}^{∞}(f̂(n) + f̂(−n)) cos(nx) + i(f̂(n) − f̂(−n)) sin(nx).

(b) Prove that if f is even, then f̂(n) = f̂(−n), and so the Fourier series is a cosine series.

(c) Prove that if f is odd, then f̂(n) = −f̂(−n), and so the Fourier series is a sine series.

(d) Suppose that f(x + π) = f(x) for all x ∈ R. Show that f̂(n) = 0 for all odd n.

(e) Show that if f is real-valued then \overline{f̂(n)} = f̂(−n). *7. Recall that the Dirichlet kernel is

D_{N}(x) = ∑_{n=−N}^{N} e^{inx}.

(a) Prove that

D_{N}(x) = \frac{sin((N + \frac{1}{2})x)}{sin(x/2)}.

(b) Show that D_{N} is an even function and that

\frac{1}{2π} ∫_{−π}^{π} D_{N}(x) dx = 1.

These exercises are taken from Stein and Shakarchi.