Math 641-600 Assignments
Assignment 1 - Due Thursday, September 6.
- Read sections 1.1-1.3.
- Do the following problems.
- Section 1.1, problem 5, page 49.
- Section 1.1 problem 9(b), page 50. (Do up to degree 3.)
- 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 if there is a
u0 for which
min u ||v-u|| = ||v-u0||,
then v-u0 is orthogonal to every u in U. Also,
show that u0 is unique, if it exists.
- Consider the tent function T(x) := (1 −
|x|)+, and let φk(x) = T(n(x-k/n)). Show that
the set B = {φk, k=0,...,n} is a basis for the space of
linear splines with nodes at x = 0,1/n,2/n,...,1.
- Section 1.2, problem 1, page 50.
- Section 1.2, problem 9, page 52.
Assignment 2 - Due Thursday, September 13.
- Read sections 1.1-1.5
- Do the following problems.
- Section 1.2, problem 5(b), page 51.
- Section 1.2 problem 6(b), page 51.
- Let U be a finite dimensional subspace of an inner product space
V, with the inner product and norm being < ·,· >
and ||·||. Define the projection of v in V onto U via Pv =
u0. Show that P is a linear transformation and
that P2 = P. (You may use the theorem we stated in class on
9/6/07, which characterizes u0 via the normal equations.)
- This problem concerns several important inequalities and the
p-norms for Rn.
- Show that if α, β are positive and α + β
=1, then for all u,v ≥ 0 we have
uαvβ ≤ αu + βv.
- Let x,y be in Rn, and let p > 1 and define
q by q-1 = 1 - p-1. Prove Hölder's
inequality,
|∑j xjyj| ≤ ||x||p
||y||q.
Hint: use the inequality in part (a), but with appropriate choices of
the parameters. For example, u =
(|xj|/||x||p)p
- Let x,y be in Rn, and let p > 1. Prove
Minkowski's inequality,
||x+y||p ≤ ||x||p + ||y||p.
Use this to show that ||x||p defines a norm on
Rn. Hint: you will need to use Hölder's
inequality, along with a trick.
- Let A be an n×n real, invertible matrix, with B =
A−1 being its inverse. Using the definition of
biorthogonal sets of vectors given in Section 1.2, problem 9, page 52,
show that the columns of A and B* (the transpose of B) are
biorthogonal.
Assignment 3 - Due Thursday, September 20.
- Read sections 1.1-1.5
- Do the following problems.
- Section 1.2, problem 10(a), page 52.
- 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.)
- Let U be a finite dimensional subspace of a finite dimensional
inner product space V, with the inner product and norm being <
·,· > and ||·||. Let Pv be the projection of v
onto U. That is, for v in V, Pv is the unique vector in U that is
closest to v; Pv satisfies the normal equations; P2 = P,
etc.
- Show that P is selfadjoint; i.e., P* = P.
- Show that if v is in U⊥, then Pv = 0.
- Find the eigenvalues and eigenvectors of P.
- 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.)
- Do an internet search for Gershgorin's theorem. Write
out both a statement and a proof of the theorem, even if all you do is
copy by hand what's on the web site. (Please reference the site used.)
Use Gershgorin's theorem to answer the questions below.
- Show that the eigenvalues of the matrix A (pg. 23) satisfy
λj ≤ 0. (The book directly shows that they are
actually negative.)
- Consider the matrix B =
4 1 1
0 3 -2
1 2 4
Show that the eigenvalues of B are all in the right-half plane. (I.e.,
Re(λ) > 0.)
- Section 1.3, problem 2, page 53.
Assignment 4 - Due Thursday, September 27.
- Read sections 2.1-2.2.2
- Do the following problems.
- Section 1.4, problem 3, page 54.
- Section 1.4, problem 4, page 54.
- Suppose data samples Y-2, Y-1,
Y0, Y1, Y2 are taken at the times t =
t0 − 2h, t0 − h, t0,
t0 + h, t0 + 2h.
- For this data, find the linear least squares fit, y(t) = a + bt.
- Show that y(t0) = (Y-2 +
Y-1 + Y0 + Y1 +
Y2)/5.
- Smoothing filter. A strategy for removing noise from
data is use the average at t0 to replace the actual data at
t0. That is, we replace Y(t0) by
y(t0). Perform the following numerical experiment. Let f(t)
= sin(πt), where −1 ≤ t ≤1. Sample f(t) at 101 equally
spaced points, starting at t = −1 and ending at t = 1. (This
means that h = 0.02). To each sample value f(t), add random noise
between −0.05 and +0.05. This gives a vector fn(t). Use the
strategy above to filter fn to get y(t). How well does y compare to
f? (In Matlab, you can use the
filter
command. But be
careful! This gives something slightly different from what you want.)
Make separate plots of f-fn and f-y vs. t.
- Section 1.5, problem 1(b), page 54.
- Section 1.5, problem 3, page 55.
- Section 1.5, problem 4, page 55.
- Section 1.5, problem 10, page 56.
Assignment 5 - Due Thursday, October 4.
- Read sections 2.1-2.2.2
- Do the following problems.
- Section 2.1, problem 5, page 94.
- Section 2.1, problem 6, page 94.
- Calculus problem: Let g be C2 on an interval
[a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
||g||C[a,b] ≤ (h2/8)
||g′′||C[a,b].
Give an example that shows
that 1/8 is the best possible constant.
- Use the previous problem to show that if f ∈
C2[0,1], then the equally spaced linear spline interpolant
fn satisfies
||f −
fn||C[a,b] ≤ (8n2) −
1 ||f′′||C[0,1]
- Fix f∈ C[0,1] and let ω(f;δ) be the
mudulus of continuity of f.
- Show that if f ∈ C1[0,1], then
ω(f;δ) ≤ δ||f′||C[0,1]
- Show that ω(f;δ) is nonincreasing as a
function of δ.
- In class, we said that lim δ↓0
ω(f;δ) = 0. Show that this is true.
- Recall that a function f:[0,1] → R is
measurable if and only if for all c ∈
R the set f −1(c,∞) is measurable.
- Show that the characteristic function of A, which is defined by
χ(t) = 1 if t ∈ A, and χ(t) = 0 if t ∉ A,
is measurable.
- Show that ∫ 01χ(t) dt = m(A).
Assignment 6 - Due Thursday, October 11.
- Read sections 2.1-2.2.2
- Do the following problems.
- Let F(s) = ∫ 0∞ e − s
t f(t)dt be the Laplace transform of f ∈
L1([0,∞)). Use the Lebesgue dominated convergence
theorem, which holds here, to show that F is continuous from the right
at s = 0. That is, show that
lims↓0 F(s) = F(0)=
∫ 0∞f(t)dt.
- 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.
- Section 2.1, problem 10, page 94.
- Section 2.1, problem 11, page 94.
- 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.
Assignment 7 - Due Thursday, November 8.
- Read sections 2.2.7, 3.2.
- Do the following problems.
- Section 2.2, problem 2(a,c), page 95. (Use the real form of the
Fourier series; see p. 75 for formulas.)
- Section 2.2, problem 9, page 95.
- Consider all 2π periodic functions f with Fourier series f(t)
= &sumn cneint such that
&sumn |cn| is convergent. Show that the
Fourier series for f converges uniformly to f and that f is continuous.
- 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 8 - Due Thursday, November 15.
- Read sections 3.2 and 3.3.
- Do the following problems.
- Section 2.2, problem 18, page 97.
- Section 2.2, problem 25, page 98.
- 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.
- Write the BVP in weak form.
- 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 is the linear B-spline. 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.
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
Assignment 9 - Due Tuesday, November 27.
- Read sections 3.3 and 3.4.
- Do the following problems.
- 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 is a closed subspace of a Hilbert space H.
-
Show that if h ∈ H and if
α := infv∈M|| h- v||,
then there is a sequnce of vectors {pj} in M for which
limj→∞|| h- pj|| = α
- In class, we used the polarization identity to show that
||pj − pk||2 =
2||h − pj||2 +
2||h − pk||2 −
4||h − ½(pj+pk)||2.
Use this identity to prove that {pj} is a Cauchy
sequence. From there, finish the proof of the Projection theorem.
- Show that M is closed if and only if M =
(M&perp)&perp.
- Weak solutions. In this problem, let
< f,g >H = ∫01(f
′(x) g ′(x) +P(x)f(x)g(x))dx,
where f,g are continuous functions vanishing at x = 0 and x = 1, and
having L2 derivatives. In addition, we assume that P is
continuous and strictly positive on [0,1]. Show that the boundary
value problem (BVP) below has a unique weak solution in H, given that
H is known to be a Hilbert space. (Hint: use the Riesz Representation
Theorem.)
-u''+P(x)u = f(x), u(0) = u(1) = 0, f ∈ L2[0,1].
- 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.)
- A sequence {fn} in H is said to be weakly
convergent to f∈H if and only if lim n →
∞ < fn,g> = < f,g> for every
g∈H. When this happens, we write f = w-lim fn. For
example, in class we showed that if {φn} is any
orthonormal sequence, then φn converges weakly to
0. One can show that every weakly convergent sequence is a bounded
sequence; that is, there is a constant C such that ||fn||
≤ C for all n. Prove the following:
Let K be a compact linear operator on a Hilbert space H. Show
that if fn weakly converges to f, then Kfn
converges strongly to Kf that is,
lim n →
∞ || Kfn - Kf || =0.
Hint: Suppose this doesn't happen, then there will be a subsequence of
{fn}, say {gk}, such that
|| Kgk - Kf || ≥ ε
for all k. Use this and the compactness of K to
arrive at a contradiction.
Updated 11/15/07 (fjn).