Homework #6

Math 115 Spring 2015
Problem Set 6
Prof. Bernoff
Orthogonal Functions, L2 Convergence and Fourier Series
Due date: Wednesday, March 4th in class.
For these problems, you may use MAPLE when appropriate.
F1: Gram-Schmidt orthogonalization. We know from elementary linear algebra that any
set of linearly independent vectors may be turned into a set of orthogonal vectors by the
Gram-Schmidt orthogonalization process. The same process works for functions defined on
an interval a < x < b (a mathematician would say on L2 [a, b]). Let {fn } be a linearly
independent set of functions. Define an inner-product
Z b
hp(x), q(x)i =
p(x) · q(x) dx
x=a
Define the sequence of functions gn is defined inductively by
g1 = f1 ,
g2 = f2 −
hf2 , g1 i
g1 ,
||g1 ||2
g3 = f3 −
hf3 , g1 i
hf3 , g2 i
g1 −
g2 , . . . .
2
||g1 ||
||g2 ||2
where ||g||2 = hg, gi. Use induction to show that gn is an orthogonal sequence.
F2: Approximating a function by Legendre Polynomials. This is a great problem for you
to hone your MAPLE skills on.
(a) The functions {1, x, x2 , x3 , . . . }, are linearly independent on the interval x ∈ [−1, 1]. Use
the previous problem and the functions {1, x, x2 , x3 } to generate a sequence of four orthogonal polynomials on x ∈ [−1, 1]. Denote the polynomials by P0 (x), P1 (x), P2 (x), P3 (x)
(they are called the Legendre Polynomials).
(b) Find an approximation of ex on x ∈ [−1, 1] of the form
ex ≈ c0 P0 (x) + c1 P1 (x) + c2 P2 (x) + c3 P3 (x)
that is best in the root-mean-square sense, and graph ex and the approximation on a
set of coordinate axes.
(c) For your approximation what is the maximum of the pointwise error on [−1, 1]? What
is the L2 error?
Editorial Note: The pointwise error of an approximation F ∗ (x) to a function F (x) is
E(x) = |F (x) − F ∗ (x)|. The root-mean-square or L2 error here is
sZ
1
[E(x)]2 dx
||E(x)|| =
−1
(please turn over)
Math 115 Spring 2015
Problem Set 6
Prof. Bernoff
F3: Consider a periodic function, g(t), with period 2π and
g(t) = t2
−π <t<π
(a) Find a Fourier cosine series for g(t). Check your answer by graphing the partial sums
with MAPLE.
(b) Differentiate the cosine series and the function to obtain a sine series for
f (t) =
1 dg
=t
2 dt
−π <t<π
Check your answer by graphing the partial sums with MAPLE. What happens
at t = ±π?
(c) Show by consider g(0) that
∞
π 2 X (−1)n+1
=
.
12 n=1
n2
What do you need to assume for this identity to be true?
(d) Prove the two identities
∞
π2 X 1
=
6
n2
n=1
∞
π4 X 1
=
90 n=1 n4
by applying Parseval’s Theorem to f (t) and g(t) respectively. What have you assumed
here and how does it differ from part (c)?
F4: Consider a function, h(x), defined for −π < x ≤ π
(
2x 0 ≤ x ≤ π
h(x) =
0
−π < x ≤ 0
(a) Draw the 2π-periodic extension of h(x).
(b) Compute the Fourier expansion for h(x),
∞
a0 X
+
an cos(nx) + bn sin(nx) .
h(x) =
2
n=1
Graph some partial sums and show that it converges to the periodic extension of h(x).
What happens at π?
(please turn over)
Math 115 Spring 2015
Problem Set 6
Prof. Bernoff
(c) Note that the even part of h(x) is given by
E(x) =
1
[h(x) + h(−x)] = |x|
2
−π ≤x≤π .
Use this identity to compute the series for E(x) = |x| on this interval. Show the
series reduces to a cosine series (why is this true?). You can verify this result with the
calculation we did in class.
(d) Note that the odd part of h(x) is given by
O(x) =
1
[h(x) − h(−x)] = x
2
−π <x<π .
Use this identity to compute the series for O(x) = x on this interval. Show the series
reduces to a sine series (why is this true?). You can verify this result with H3(b).