Name:
CSU ID:
Homework 11
April 24, 2015
1. S6.3 ]2(b),(d)
2. S6.3 ]4
3. S6.3 ]10,14
HW12 will include the following problems.
1. S6.3 ]22
2. S6.3 ]26
3. S6.3 ]34
4. S6.3 ]43
5. S6.3 ]44
6. (It is likely this problem will be on HW 12) Let V be the space of
piecewise continuous functions on −π
≤ x ≤ π. For functions u, v ∈ V ,
Rπ
let the inner product be hu, vi = −π u(x)v(x)dx. Consider the basis
functions on [−π, π] defined by
1
1
g0 (x) = √ , gk (x) = √ cos(kx)
π
2π
and
1
fk (x) = √ sin(kx)
π
(a) Show that the basis functions are orthonormal. Use integral tables.
(b) Find the projection of F (x) = x − 1 onto g0 , g1 , g2 , f1 , f2 . Use
integral tables.
(c) Find the seond order Fourier expansion for F (x) = x − 1. Use
integral tables.