Math 641-600 Fall 2011
Assignments
Assignment 1 - Due Monday, 9/5/2011
- Read sections 1.1-1.3.
- Do the following problems.
- Section 1.2, problem 1, page 50.
- Section 1.2, problem 6(d), page 51.
- Section 1.2, problem 9, page 52.
- Let V be a finite dimensional inner product space and let W be a
subspace of V. The orthogonal complement of W is
W⊥ = {v ∈ V | < v,w> = 0 for
all w ∈ W}
Show that V = W⊕W⊥, where ⊕ symbolizes
the direct sum of vector spaces. Also, show that
(W⊥)⊥ = W.
Assignment 2 - Due Wednesday, 9/14/2011
- Read sections 1.4, 2.1
- Do the following problems.
- Section 1.3: problem 2
- Section 1.3: problem 5
- Section 1.4: problem 3
- Section 1.4: problem 4
- Let Q be a unitary, n×n matrix. Do the following.
- Show that < Qx, Qy > = < x,
y >.
- Show that the eigenvalues of Q all lie on the unit circle,
|λ|=1.
- Show that Q is diagonalizable. (Hint: Modify the proof that a
self adjoint matrix is diagonalizable; this starts in the middle of
p. 15 of the text. To prove invariance, you will need to make use of
Q* = Q −1.)
- This problem concerns a second proof that a self-adjoint matrix A
is diagonalizable. You may use the fact that the eigenvalues of A are
real
- Show that A has at least one eigenvalue λ1,
with a corresponding eigenvector ψ1.
- Select an orthonormal basis, {ψ1,
φ2,
, φn}. Let Q1 =
[ψ1 φ2
φn]. Show
that Q*1A Q1 is a matrix with
(1,1)-entry equal to λ1, all other entries in the
first row and column are 0, and the remaining
n−1×n−1 matrix A1 is self adjoint.
- Repeat the procedure to find a unitary matrix Q that diagonalizes
A. (Q will be a product of the unitary matrices used at each step.)
Assignment 3 - Due Friday, 9/23/2011
- Read sections 2.1-2.2.2.
- Do the following problems.
- This problem concerns several important inequalities.
- 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 ∈ 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 ∈ 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.
- Show that ℓ2 is a Hilbert space -- i.e., that
it's complete -- under the inner product ⟨x,y⟩ =
∑j
xjyj. (j = 0,...)
- Let f(x) : x2, -1 ≤ x ≤ 2. Find f
−1(Ej) for Ej = [j/2, (j+1)/2), j
= 0,
, 7. Using these, find the numerical value of
the Lebesgue sum corresponding to y*j = (2j+1)/4.
- Section 2.1, problem 3, page 94.
- Section 2.1, problem 5, page 94.
- Section 2.1, problem 8, page 94.
Assignment 4 - Due Monday, 10/3/2011
- Read sections 2.2.3, 2.2.4, 2.2.7.
- Do the following problems.
- Section 2.1, problem 9, page 94.
- Section 2.1, problem 10, page 94.
- Section 2.2, problem 8, page 96.
- Prove the following proposition:
Let V be a normed linear space, with ||·|| being the norm. If
{vn} is a Cauchy sequence in V, then there is a
constant C > 0, independent of n, for which
||vn|| < C for all n.
- Let F(s) = ∫ 0∞ e − s
t f(t)dt be the Laplace transform of f ∈
L1([0,∞)). Use the Lebesgue dominated convergence
theorem 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
- Fix f ∈ C[0,1] and let ω(f;δ) be the
modulus of continuity of f.
- Fix δ. Let Sδ = { ε > 0 | |f(t)
− f(s)| < ε for all s,t ∈ [0,1], |s − t|
≤ δ}. In other words, for given δ, ε is in the
set if |f(t) − f(s)| < ε holds for all |s − t|
≤ δ. Show that
ω(f;δ) = inf Sδ
- 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.)
- Show that lim δ↓0 ω(f;δ) = 0.
- Find the best least-squares approximation in L2[-1,1]
to ex from span{1,x,x2}. Use the first three
Legendre polynomials to do this. Make a plot of ex and
the quadratic polynomial that best fits ex.
Assignment 5 - Due Friday, 10/14/2011
- Read sections 2.2.3, 2.2.4 and Notes on the Discrete Fourier Transform, 2.2.7.
- Do the following problems.
- Section 2.2, problem 9, page 96.
- Section 2.2, problem 14, page 97.
- Let w(x) ∈ C[0,1] be strictly positive on [0,1]. Show that
the orthogonal polynomials obtained via Gram-Schmidt, with the inner
product being
< f,g > =
∫01f(x)g(x)w(x)dx,
form a complete orthogonal set relative to this inner product.
- Compute the Fourier series for the following functions. For each
of these, write out the corresponding version of Parseval's identity. In each case, estimate the L2 error made when the series is truncated at a large integer N.
- f(x) = x, 0≤ x ≤ 2π
- f(x) = |x|, − π ≤ x ≤ π
- f(x) = e2x, 0≤ x ≤ 2π (complex form).
- Consider the series ∑n cn einx, where ∑n |cn| < ∞. Show that ∑n cn einx converges uniformly to a continuous function f(x) and that the series is the Fourier series for f.
- Prove the Convolution Theorem for the
DFT. (See
Notes on the Discrete Fourier Transform, pg. 3.)
Assignment 6 - Due Friday, 11/4/2011
- Read sections 2.2.7, 3.2, 3.3
- Do the following problems.
- Section 2.2, problem 26(a), page 99.
- Section 2.2, problem 27(a), page 99. Hint for 27(a): Use the
normal equations show that the interpolant f ∈
S1/n(3,1) that minimizes
∫01 (f ′′(x))2dx
satisfies ∫01 f ′′(x)
ψ′′j(x) dx = 0. Second, show that, for any
cubic polynomial p on [a,b] and any function h ∈
C(2)[a,b], one has
∫a1 p ′′(x) h ′′(x)
dx = p′′(x)h′(x) - p′′′
h(x)|ab. (Note: p′′′ is
constant on [a,b].) Applying these appropriately will solve the
problem.
- Let V be a real Banach space and let V* be the dual space of
V. Show that V* is a Banach space with the norm ||Φ|| =
sup{|Φ(v)| : ||v||=1}. (This is the operator norm for Φ : V
→ R.)
- Let M be a subspace of a Hilbert space H. Show that M is closed
if and only if M = (M⊥)⊥.
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous. We say that L is continuous
at v ∈ V if and only if
for every ε > 0 there is a δ > 0 such that ||L(v)
− L(u)|| < ε whenever ||v - u|| < δ. For a linear
operator, it is only necessary to show continuity at v = 0.
- 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. Recall that the operator P:H → H by Ph = p is called
the projection of H onto M, and that P2 = P. Show
that the following are true.
- P is a bounded linear operator, with ||P||=1.
- Range(P) = M and Null(P) = M⊥.
- P is self-adjoint, i.e. P* = P.
Assignment 7 - Due Monday, 11/14/2011
- Read sections 3.3-3.5
- Do the following problems.
- Section 3.2: problem 3(b,d), page 128.
- Let L be a bounded linear operator on Hilbert space H. Show that
the two formulas for ||L|| are equivalent:
- ||L|| = sup {||Lu|| : u ∈ H, ||u|| = 1}
- ||L|| = sup {|< Lu,v >| : u,v ∈ H, ||u|| = 1 and ||v||
= 1}
- Consider the finite rank (degenerate) kernel k(x,y) =
φ1(x)ψ1(y) +
φ2(x)ψ2(y),
where φ1 = 2x-1, φ2 = x2,
ψ1 = 1, ψ2 = x.
-
For what values of λ does the integral equation
u(x) = f(x) + λ∫01 k(x,y)u(y)dy
have a solution for all f ∈ L2[0,1]? For
these values, find the solution u = (I −
λK)−1f i.e., find the resolvent. Here,
Ku(x) = ∫01 k(x,y)u(y)dy.
- For the values of λ for which the equation
does not have a solution for all f, find a condition on f
that guarantees a solution exists. Will the solution be unique?
- (Cea's Lemma) In this problem, let H1[0,1] be the
usual Sobolev space, H={u∈ H1[0,1] : u(0)=u(1)=0}, and
let V(x) be continuous and strictly positive on [0,1]. You are given
that H is a (real) Hilbert space relative to the inner product
< u, v >H =
∫01(u′(x)v′(x) + V(x)u(x)v(x))dx,
as well as relative to the usual inner product on H1.
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.
We
want to approximate the unique weak solution u to −u''+V(x)u =
h(x), u ∈ H, h ∈ L2, which solves
the problem
< u, f >H = ∫01h(x)f(x)dx
∀ f ∈ H.
To do that, we use functions from a finite dimensional subspace M of
H. Show that the solution uM to the problem for M,
< uM, f >H =
∫01h(x)f(x)dx ∀ f ∈ M,
is nearly a best approximate to u in the norm of
H1[0,1]. That is, if vM ∈ M satisfies
||u − vM||H1 = min{|| u−
v||H1 ∀ v ∈ M},
then there is a constant C such that
C ||u − uM||H1 ≤ ||u −
vM||H1 ≤ ||u −
uM||H1.
- A sequence {fn} in a Hilbert space 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.
Assignment 8 - Due Wednesday, 11/23/2011
- Read sections 3.6, 4.1
- Do the following problems.
- Section 3.3: problem 1, page 128.
- Section 3.4: problem 2(b,d), page 129. (Hint for part (d):
Convert the integral equation into a boundary value problem for an
ordinary differential equation. Proceed as in example 2, p. 118.)
- Section 3.4: problem 3. (Hint: repeat the procedure used in
3.4.2(d) in the previous problem. Show that this time the boundary
conditions for u are u(1) = −u(−1) and
u′(1) = −u′(−1).)
- Section 3.5: problem 1(b), page 129.
- Section 3.5: problem 2(a), page 129.
- Let K be a compact, self-adjoint operator. Show that the only
possible limit point of the set of eigenvalues {μj} of K
is 0 i.e., the non-zero eigenvalues of K are isolated.
- Let K be a compact, self-adjoint operator. Show that the set of
orthonormal eigenfunctions {φj} corresponding to
μj ≠ 0 is complete in H if and only if 0
is not an eigenvalue of K.
Assignment 9 - Due Friday, 12/2/2011
- Read sections 4.1,4.2
- Do the following problems.
- Section 3.6: problem 1, page 130.
- Section 3.6: problem 4, page 131.
- Section 3.6: problem 5, page 131.
- Consider eigenvalue problem y′′ = - λ y. with
boundary conditions y(0)=0, y(1)+y′(1) = 0. Show that the
eigenvalues satisfy the equation tan(λ1/2) +
λ1/2 = 0 and the eigenfunctions have the form
φ(x) = sin(λ1/2x). Show that the
eigenfunctions are a complete set in L2[0,1] by
converting the problem to one of the form
λ−1 u = Ku for an appropriate kernel. (Hint:
to convert the problem, use variation of parameters.)
- Section 4.1: Problem 1(c), page 171.
Updated 11/23/2011.