Math 641-600 Fall 2010
Assignments
Assignment 1 - Due Wednesday, 9/8/2010.
- Read sections 1.1-1.4.
- Do the following problems.
- Section 1.1: 5, 9(a). (In 9(a), do only the polynomials up to
degree 2.)
- Let U be a subspace of an inner product space V, with the inner
product and norm being < ·,· > and ||·||
Also, let v ∈ V. (Do not assume that U is finite
dimensional or use arguments requiring a basis.)
- Show that there is a p for which
minu ∈ U ||v − u|| =
||v − p||,
if and only if v − p is orthogonal
to the subspace U.
- Show that p is unique, given that it exists
for v.
- Suppose p exists for every v ∈
V. Since p is uniquely determined by v, we may define
a map P: V → U via Pv := p. Show that P is a
linear map and that P satisfies P2 = P. (P is
called an orthogonal projection. The vector p is the
orthogonal projection of v onto U.)
- In the previous problem, let U be finite dimensional, with a
basis B = {u1, ..., un}.
- Show that the n×n matrix G, with entries Gj,k =
<uk,uj>, is invertible. (G
is called a Gram matrix for the basis B.)
- Show that for every v ∈ V, the orthogonal projection
of v onto U is given by
p = ∑k xkuk, where
the xj's satisfy the system
<v,uk > = ∑j
Gk,jxj. That is, if we let dk :=
<v,uk >, then d = Gx.
- Show that if B is orthonormal, p = ∑k
<v,uk >uk.
- Use your answer to section 1.1, 9(a) to find the best quadratic
fit to e−x relative the inner product in that problem.
- 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.
Assignment 2 - Due Wednesday, 9/15/10.
- Read sections 2.1-2.2.2.
- Do the following problems.
- Section 1.3: 2, 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⊥. 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. (Hint: Modify the proof that a
self adjoint matix is diagonalizable; this starts in the middle of
p. 15 of the text.)
- 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?
Assignment 3 - Due Thursday, 9/23/10.
- Read sections 2.1-2.2.2.
- Do the following problems.
- Section 2.1: 3, 5, 10
- 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.
- 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.
- Let f ∈ C(2)[0,1]. Show that if fn is
the equally-spaced interpolating spline for f, then
||f − fn||C[0,1] ≤ ½
n−2||f′′||C[0,1].
Hint: first show this: If a < b, g ∈ C(2)[a,b], and
g(a) = g(b) = 0, then ||g||C[a,b] ≤
½(b-a)2||g′′||C[a,b].
- 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}.
Assignment 4 - Due Wednesday, 10/6/10.
- Read sections 2.2.1-2.2.3.
- Do the following problems.
- Section 2.1: 9
- Section 2.2: 7, 8
- 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
- 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.
- 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.
- f(x) = x, 0≤ x ≤ 2π (π = "pi".)
- 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.
Assignment 5 - Due Friday, October 15.
- Read sections 2.2.4, 2.2.6 (B-splines), and 2.2.7. 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.
- Do the following problems.
- Section 2.2: 25, 26(a), 27(a)
- 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, respectively.
-
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.
- 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. Also, et H have the
inner product:
⟨f,g⟩H = ∫01 f
′(x) g ′(x) dx.
Use integration by parts to convert the BVP to its weak form:
⟨u,v⟩H=∫01 f(x)
v(x) dx for all v ∈ H.
- Let V = Sh(1,0), with h = 1/n. Thus V is spanned by
φj(x) := N2(nx-j+1), j = 1 ...
n-1. (Here, N2(x) is defined on p. 81.) Show that the
least-squares approximation to u from V is y = ∑j
αjφj(x) ∈ V, where the
αj's satisfy Φα = β, with
βj = ⟨ y,φj ⟩H
= ∫01 f(x) φj(x) dx, j=1
... n-1 and Φkj = ⟨ φj,
φk ⟩H.
- Show that Φkj = ⟨ φj,
φk ⟩H is given by
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.
Write out this matrix explicity when n = 6.
Assignment 6 - Due Wednesday, November 3.
- Read sections 3.1-3.3.
- Do the following problems.
- 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.
- Consider space H = comprising all functions f in Sobolev space
H1[0,1] that satsify f(0) = f(1) = 0. Let V(x) 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.
Assignment 7 - Due Friday, November 12.
- Read sections 3.1-3.3.
- Do the following problems.
- Section 3.2: 3(b,d)
- 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, use the notation, assumptions and
conclusions of problem 5, Assignment 6. In particular,
H1[0,1] is the usual Sobolev space and H={u∈
H1[0,1] : u(0)=u(1)=0}. Also, 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. 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, December 1.
- Read sections 3.4-3.6, 4.1.
- Do the following problems.
- Section 3.4: 1(a,b), 2
- Section 3.5: 1(b), 2(b)
- Section 3.6: 5
- Section 4.1: 4
- 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., μ ≠0 is never a limit point of the set of eigenvalues.
Updated 11/18/2010.