hw30
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moienMath 21b: Linear Algebra Spring 2016
Homework 30: Fourier II
This homework is due on Friday, April 22, respectively on Tuesday, April 26, 2016.
1 Find the Fourier series of the function which is 1 on [0, π/2] and
zero everywhere else.
Z π
2 Use Parseval to find −π f (x)2 dx for
f (x) = cos(11x)+cos(13x)+2 sin(17x)−cos(19x)+5 cos(1111x)
3 Compute both sides of the Parseval identity for f (x) = x + |x|.
4 Find ∞ 1
n=1 (2n)2 = 1/4 + 1/16 + 1/36 + ... from the known formula
P
of n n12 and use this to compute the sum ∞ 1
n=0 (2n+1)2 over the
P P
odd numbers.
5 This problem is a preparation for PDEs and consists of reminders.
All statements are pretty straight forward if we work with func-
tions on [−π, π] described by Fourier series:
a) Verify that the Fourier basis B = {1, cos(nx), sin(nx)} consists
of eigenfunctions of D2.
b) What are the corresponding eigenvalues?
c) Show that every eigenfunction of D2 is either constant or of the
form a cos(nx) + b sin(nx) for some n.
d) What are the eigenvalues of D2 + D4 + 6 on the subspace of
∞
Cper consisting of odd functions?
Fourier Series II
∞
Recall that the Fourier coefficients of a function f ∈ Cper
√ √
are defined as a0 = hf, 1/ 2i = π1 −π
Rπ
f (x)/ 2 dx, an =
hf, cos(nt)i = π1 −π
Rπ
f (x) cos(nx) dx, bn = hf, sin(nt)i =
1 Rπ
π −π f (x) sin(nx) dx. These are just inner products an =
hf, cos(nx)i and bn = hf, sin(nx)i and
√
f = a0/ 2 + an cos(nx) + bn sin(nx)
X X
n n
is the Fourier series of f . The Parseval identity is ||f ||2 =
a20 + ∞ 2 2
k=1 ak + bk . It is an extension of the Pythagoras theorem.
P