Math 118 :: Winter 2009 :: Homework 1 :: Due January 22
1. Prove the following properties of the Fourier transform (x, k ∈ R):
(a) Dilation: if g(x) = f (x/a) for a > 0, then ĝ(k) = afˆ(ak).
(b) Conjugation: if g(x) = f (x), then ĝ(k) = fˆ(−k).
(c) Symmetry: if f is real and even (f (x) = f (−x)), show that fˆ(k) is also real and even.
(d) Symmetry: if f is real and odd (f (x) = f (−x)), show that fˆ(k) is imaginary and odd.
2. Consider the sinc function
sinc(x) =
sin x
.
x
(a) RShow that sinc is not integrable,
i.e. sinc ∈
/ L1 (R). [Hint: if f (x) ≥ g(x) ≥ 0 and
R
g(x)dx diverges, then f (x)dx diverges as well.]
(b) Nevertheless, integrals involving sinc may make sense. From the theory of Fourier transforms, predict the value of
Z
∞
sinc(x)dx.
−∞
3. Consider a kernel K(x), and the integral equation
Z ∞
u(x) +
K(x − y)u(y)dy = f (x).
−∞
Similar looking equations arise for instance in rendering in computer graphics, and in the
scattering of radar waves off of planes. Find a formula for the solution u(x) of the above
equation, using the Fourier transform. What is the condition on K(x) such that no division
by zero occurs?
4. Consider a differentiable function f (x) defined on [−π, π], and the function u implicitly defined
by
du
(x) = f (x).
dx
We are looking for periodic solutions of this equation, i.e., u(−π) = u(π).
(a) Provided u is periodic, how is the derivative du/dx computed by means of Fourier series
(FS)? Use the definition
Z π
ˆ
fk =
e−ikx f (x)dx
−π
for Fourier series.
(b) What is the condition on fˆk such that the equation has a solution u? When this condition
is satisfied, is the solution u unique or not?
(c) Find the Fourier series ĥk of the constant function h(x) = 1, x ∈ [−π, π].
(d) Of course an explicit formula for u is
Z
u(x) =
x
f (y)dy + c,
0
for some arbitrary constant c. How do you reconcile the indeterminacy of the constant
with your answers to points (b) and (c)?
Rx
(e) The function f does not need to satisfy the condition of point (b) in order for 0 f (y)dy+c
to make sense. How do you explain that this condition is nevertheless needed in point
(b)?
5. Show that the functions
vk (x) = ce−ikx ,
k∈Z
Rπ
are orthogonal over [−π, π], for the inner product hf, gi = −π f (x)g(x)dx. Find the value of
c such that these functions are also normalized. [This goes part of the way towards justifying
the formulas for the semidiscrete Fourier transform, and Fourier series.]
6. In practice it is important to have at our disposal an algorithm for the FFT for general
values of N . If 3 divides N , express the discrete Fourier transform (DFT) of length N as a
function of three DFT of length N/3. The preferred answer to this question is expository,
with sentences in addition to pictures and formulas.