Math 641-600 Assignments
Assignment 1 Due Friday, 9/11/09.
- Read sections 1.1-1.4.
- Do the following problems.
- Section 1.1: 4, 5, 7, 9(c). (In 9(c), do only the first 3
polynomials.)
- Let U be a subspace of an inner product space V, with the inner
product and norm being < ·,· > and ||·||
Also, let v be in V, but suppose v is not in U. Show that there is a
u0 for which
min u ||v-u|| = ||v-u0||,
if and only if v-u0 is orthogonal to every u in U. Also,
show that u0 is unique, if it exists.
- Section 1.2: 6(a,c), 8, 9.
- Section 1.3: 2.
Assignment 2 - Due Wednesday, 9/23/09.
- Read sections 2.1-2.2.2.
- Do the following problems.
- Section 1.3: 3, 4, 5.
- Section 1.4: 3, 4.
- Let V be a finite dimensional inner product space and let U be a
subspace of V. The orthogonal complement of U is
U⊥ = {v ∈ V | < v,w> = 0 for
all w ∈ U}
Show that V = U⊕U⊥. (You may need to look up the
definition of ⊕, which symbolizes the direct sum of
vector spaces.) Also, show that (U⊥)⊥ = U.
- Let U be a unitary, n×n matrix. Do the following.
- Show that < Ux, Uy > = < x,
y >.
- Show that the eigenvalues of U all lie on the unit circle, |λ|=1.
- Show that U is diagonalizable. (Scalars must be complex).
- Suppose that U is real as well as unitary. (Such matrices are
orthogonal.) In an odd dimensional space, show that either 1
or − 1 is an eigenvalue of U. (It's possible for both 1
and − 1 to be eigenvalues.)
- Consider a spring system with 3 equal masses (mj = m),
and 3 equal springs (kj = k) connecting them, arranged on a
circle. Here, the displacement uj is an angle
θj, and the third spring is connected to the first
mass. (Think of beads on a necklace.) Write down the equations of
motion and find the normal modes for the system. One of the normal
frequencies is ω = 0. What motion does this correspond to? What
happens to the other normal frequencies if m is fixed and k increases?
decreases?
Assignment 3 - Due Wednesday, 9/30/09.
- Read sections 2.1-2.2.2.
- Do the following problems.
- Show that ℓ2 is a Hilbert space -- i.e., that
it's complete -- under the inner product ⟨x,y⟩ =
∑j
xjyj. (j = 0,...)
- Fix f∈ C[0,1] and let ω(f;δ) be the
modulus of continuity of f.
- Show that ω(f;δ) = inf{ ε > 0 | for all
t,s in [0,1] satisfying |t − s| ≤ δ one has |f(t)
− f(s)| < ε}.
- Show that ω(f;δ) is non decreasing as a
function of δ. (Or, more to the point, as δ ↓ 0,
ω(f;δ) gets smaller.)
- In class, we said that lim δ↓0
ω(f;δ) = 0. Show that this is true.
- Find two Lebesgue sums approximating the integral of f(x) =
x2, −1 ≤ x ≤ 2, given the y-axis partition
{0, 1/2, 1, 3/2, 2, 3, 4}.
- Assume that if A ⊂ [0,1], then m∗(A) ≤
m∗(A). Use this to show that if B ⊂ A ⊂
[0,1] and if m(A) = 0, then B is measurable and m(B) = 0.
Assignment 4 - Due Friday, October 9, 2009.
- Read sections 2.2.1-2.2.3.
- Do the following problems.
- Let fn(x) = n3/2 x e-n x, where
x ∈ [0,1] and n = 1, 2, 3, ....
- Verify that the pointwise limit of fn(x) is f(x) = 0.
- Show that ||fn||C[0,1] → ∞ as n
→ ∞, so that fn does not converge uniformly to
0.
- Find a constant C such that for all n and x fixed
fn(x) ≤ C x-1/2, x ∈ (0,1].
- Use the Lebesgue dominated convergence theorem to show that
limn→ ∞ ∫01
fn(x)dx = 0.
- Find the best least-squares approximation in L2[0,1]
to ex from span{1,x}. Make a plot of ex and
the line that best fits ex.
- Section 2.2: 8.
- Let f ∈ C1[0,1], and suppose that
f(0) = f(1) = 0. Show that this version of the Sobolev inequality
holds for all x ∈ [0,1]:
|f(x)| ≤
(∫01|f′(t)|2 dt)1/2.
- Variational/Finite-element problem. We want to solve the boundary value
problem (BVP): −u'' = f(x), u(0) = u(1) = 0.
- Let H be the set of all
continuous functions vanishing at x = 0 and x = 1, and
having L2 derivatives. Show that
⟨f,g⟩H = ∫01 f ′(x) g
′(x) dx,
is an inner product for H. (Hint: To show positivity, use the previous
problem.)
- Let f(x) = x2 in the rest of the problem. Find the
exact solution to the BVP for this choice of f.
- Let φj(x) := N2(nx-j+1), where
N2(x) is the linear B-spline, or ``tent'' function,
defined on pg. 81. Find the βj's,
where
βj = ⟨ y,φj ⟩H
= ∫01 f(x)
φj(x) dx, j=1 ... n-1.
- Show that Φkj = ⟨ φj,
φk ⟩H, the k-j entry in the Gram
matrix for the problem, satisfies
Φj,j = 2n, j = 1 ... n-1
Φj,j-1 = - n, j = 2 ... n-1
Φj,j+1 = - n, j = 1 ... n-2
Φj,k = 0, all other possible k.
For example, if n=5, then Φ is
10 -5 0 0
-5 10 -5 0
0 -5 10 -5
0 0 -5 10
- Numerically solve Φα = β for n = 10, 25, 50. Use
your favorite software (mine is MATLAB) to plot the exact solution y
and, for each n, the linear finite element approximation to y,
v(x) = ∑j αj φj(x),
which is also the least squares approximation to y in the inner
product ⟨ ·,· ⟩H defined above.
- Bonus. Section 2.2: 4.
Assignment 5 - Due Monday, November 9, 2009.
- Read sections 2.2.6, 2.2.7, 3.1.
- Do the following problems.
- Compute the Fourier series for the following functions. For each
of these, write out the corresponding version of Parseval's identity.
- f(x) = x, 0≤ x ≤ 2π
- f(x) = |x|, − π ≤ x ≤ π
- f(x) = e2x, 0≤ x ≤ 2π (complex form).
- f(x) = 1 on |x| ≤ ¼ π and f(x) = 0 for all
other x in − π ≤ x ≤ π.
- Let fN be the partial sum for the series in (b)
above. Estimate the error ||f - fN|| when the norm is for
L2 and when the norm C[− π,&pi:]. (Hint: use the
esitmate form the integral test.)
- Consider a 2π periodic function f with Fourier series f(t) =
∑n cneint. Show that if f is
C(k), then |cn| ≤ C |n|− k
for all n ≠ 0.
- Prove the Convolution Theorem for
the DFT. (See Notes on the
Discrete Fourier Transform, pg. 3.)
- Let a, x, y be column vectors with entries a0, ...,
an-1, etc., and let α,ξ,η be n-periodic
sequences, with the entries for one period being those of a, x,
and y, repectively.
-
Show that the convolution η = α∗ξ is
equivalent to the matrix equation y=Ax, where A is an
n×n matrix whose first column is a, and whose remaining columns
are a with the entries cyclically permuted. For example, if n = 4, and
a = (a b c d)T, then A =
a d c b
b a d c
c b a d
d c b a
-
Such matrices are called cyclic. Use the DFT and the convolution
theorem to find the eigenvalues of a cyclic matrix A.
- Let ψ(x) be the Haar wavelet. Show that if f ∈
L2(R) is a uniformly continuous function on
R, then the wavelet coefficient
djk = 2j < f(x),
ψ(2jx - k) >
satisfies the bound |
djk | ≤ 2-1
ω(f,2-j-1), where ω is the modulus of
continuity.
- Working in the Haar MRA, where the scaling function is
φ(x)=N1(x), let f1 ∈
S1 be defined by
Express f1 in terms of the φ(2x-k)'s, then decompose
f1 into f0 ∈ S0 plus
w0 ∈ W0. Sketch all three functions.
Assignment 6 - Due Monday, December 2, 2009.
- Do the following problems.
- Section 2.2: 26. (The definition of projection applies to to
Banach spaces as well as Hilbert spaces. The two parts of the
problem involve Banach spaces.)
- Prove Proposition 0.1
in Notes
on Daubechies' Wavelets
- Prove the polarization identity for u,v in a Hilbert space H:
||u+v||2 + ||u-v||2 = 2(||u||2 +
||v||2).
- Let M be a subspace of a Hilbert space H. Show that M is closed
if and only if M = (M&perp)&perp.
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
- Let M be a closed subspace of a Hilbert space H. Let h be in H,
and let p be the unique minimizer of || h - u|| over all u in
M. Define the operator P:H → H by Ph = p. The operator P is
called the projection of H onto M. Show that the following
are true.
- P is a bounded linear operator, with ||P||=1.
- P2 = P.
- Range(P) = M and Null(P) = M⊥.
- P is self-adjoint, i.e. P* = P.
- Consider space H = comprising all functions f in Sobolev space
H1[0,1] that satsify f(0) = f(1) = 0. Let v(v) be
continuous and strictly positive on [0,1]. On H, define the inner
product
< f,g >H = ∫01(f
′(x) g ′(x) +v(x)f(x)g(x))dx,
A weak solution u to the boundary value problem (BVP)
-u''+v(x)u = h(x), u(0) = u(1) = 0, h ∈
L2[0,1].
is a function u ∈ H such that for all
f ∈ H we have < u, f >H =
∫01h(x)f(x)dx.
- Show that in the inner product above, H is a Hilbert space.
- Show that the BVP has a unique weak solution in H.
- Section 3.2 problem 3(d), page 128. (Assume the appropriate
operators are closed and that λ is real.)
- Section 3.3 problem 2, page 129. (Assume the appropriate
operators are closed and that λ is real.)
Updated 11/20/09 (fjn).