Math 641-600 Assignments
Assignment 1 - Due Wednesday, September 7.
- Read sections 2.1 and 2.2.
- Do the following problems.
- Let U be a subspace of an inner product space V, with the inner
product and norm being < ·,· > and ||·||
Also, let v ∈ V, but suppose v is not in U. Show that if there is a
u0 for which
minu ∈ U ||v-u|| = ||v-u0||,
then v-u0 is orthogonal to every u ∈ U. Also,
show that u0 is unique, if it exists.
- Review the Gram-Schmidt procedure. Do problem 6(b), page 47.
- 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.
- Problem 2, page 94.
- Problem 5, page 94.
Assignment 2 - Due Wednesday, September 14.
- Read section 2.2.
- Do the following problems.
- Problem 8, page 94 (§ 2.1). (You may use the results of
Problem 7.)
- Problem 10, page 94 (§ 2.1). Why does the Lebesgue Dominated
Convergence Theorem fail here?
- Problem 11, page 94 (§ 2.1).
- 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) = |π - x|, 0≤ x ≤ 2π
- f(x) = |x|, − π ≤ x ≤ π
- f(x) = e2x, 0≤ x ≤ 2π (complex form).
- f(x) = 1 on |x| ≤ ½ π and f(x) = 0 for all
other x in − π ≤ x ≤ π.
- Problem 12, page 97 (§ 2.2).
Assignment 3 - Due Wednesday, September 21.
- Read section 2.2.
- 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 a sequence of functions fn(t) on [0,1] such that
fn(t) → 0 as n → ∞, both pointwise and in
L1, but does not converge in L2 or C[0,1].
- Problem 1, page 95 (§ 2.2).
- Problem 3(a), page 96 (§ 2.2). (Take the degree n =3, rather
than 5.)
- Problem 5, page 96 (§ 2.2).
- Bonus. Problem 2, page 96 (§2.2).
Assignment 4 - Due Wednesday, September 28.
- Read section 2.2.
- Do the following problems.
- Finite element problem. In this problem, let
< f,g > = ∫01 f ′(x) g
′(x) dx,
where f,g are continuous, piecewise C1 functions vanishing at
x = 0 and x = 1.
-
If f(x) = x2, find the exact solution to the problem -y'' =
f(x), y(0) = y(1) = 0.
- Recall that the basis element φj(x) is the "text
function" that is defined as
φj(x) := 0 if x ≤ (j-1)/n or x ≥ (j+1)/n,
φj(x) := 1 - n|x - j/n|.
Find the βj's, where
βj = < y,φj >
= ∫01 f(x)
φj(x) dx, j=1 ... n-1.
- Show that Φkj = < φj,
φk >, 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 Φ α = 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 < · , · > defined above.
- Let f(x) be a 2π-periodic, continuous, piecewise C1
function. Show that the Fourier coefficients of f are absolutely
summable, i.e., ∑n |an| + |bn|
< ∞. (The complex version is ∑n
|cn| < ∞. It is probably easier to verify.)
Explain why this guarantees that the Fourier series for f
converges uniformly to f on, say, [0,2π].
- Problem 3(b,c), page 96 (§ 2.2). (As we did in the previous
assignment, take the degree n = 3, rather than 5.)
- Problem 4, page 96 (§ 2.2). Only do the
H1[0,2π] case. You may assume that the
integration-by-parts formula below is true for f in H1[a,b]
and g in C1[a,b]. Also, note that functions in
H1[a,b] are continuous.
∫ab f ′(x)g(x)dx = f(b)g(b) - f(a)g(a)
- ∫ab f(x)g′(x)
- Problem 6(a,b), page 96 (§ 2.2).
Assignment 5 - Due Wednesday, October 5.
- Read section 2.2.
- Do the following problems.
- Let f ∈ L2w[0,1], where w is a weight
function that is strictly positive and continuous on 0 < x ≤ 1,
and that satisfies ∫01w(x)dx = 1. (For
example, w(x) = ½ x-½ is such a function, and
so is w(x) = (3/2)x½.) Our aim is to prove the
theorem below in several steps. You may assume that all functions are
real valued.
Theorem. C[0,1] is dense in
L2w[0,1]
- Let 0 < a < 1. Show that C[a,1] is dense in
L2w[a,1], given that C[a,1] is dense in
L2[a,1]. (See pg. 65, property 7.)
- Let a/2 < δ < a < 1. Let g be continuous on
[a,1]. Extend g to be continuous on [0,1] by letting g be 0 on
[0,δ], and the line y = (g(a)/(a - δ))(x - δ) on
[δ,a]. Show that
∫0ag(x)2w(x)dx ≤
g(a)2 ∫δaw(x)dx .
- Show that for f ∈ L2w[0,1] and g as
defined above, we have
∫01(f(x) - g(x))2w(x)dx ≤
2∫0af(x)2w(x)dx +
2g(a)2 ∫δaw(x)dx
+ ∫a1(f(x) - g(x))2w(x)dx .
- Prove the theorem; that is, given ε > 0, appropriately
choose a, g, and δ, in that order, to get ||f - g||w
< ε .
- Problem 7, page 96 (§ 2.2).
- Problem 8, page 96 (§ 2.2).
- Prove the three statements made at the end of §2, p. 3 of
my Notes
on the Discrete Fourier Transform,
- Problem 15(b), page 97 (§ 2.2).
Assignment 6 - Due Wednesday, October 12.
- Read section 3.1, 3.2.
- Do the following problems.
- Let a,x,y be n-periodic sequences, and let A, X, Y be column
vectors with entries a0, ..., an-1, etc. Show
that the convolution y = a∗x is equivalent to the matrix
equation Y = αX, where α 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 α =
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 α.
- Let f be in S(3,2), the space of C2 piecewise cubic
splines on [0,1] with knots at xj = jh, h=1/N.
-
For any j, 1 ≤ j ≤ N-1, let q and Q be the cubic polynomials
that f agrees with to the left of xj and the right of
xj. Specifically, these are defined by q(x) = f(x) if
xj-1 ≤ x ≤ xj and Q(x) = f(x) if
xj ≤ x ≤ xj+1. Show that as cubic
polyomials, q(x) - Q(x) = Aj (x -
xj)3, where Aj is a constant
independent of x.
- Let 2 < j < N-2. Find S(x) ∈ S(3,2) such that
S(xj) = 1 and S(x) is 0 if x ≤ xj-2 or if x
≥ xj+2. Your answer has a very simple form if you use
the function (·)+, which we defined in class.
- Problem 24(a,d), page 99 (§ 2.2).
- Let ψ(x) be the Haar wavelet. Show that if f ∈
L2(R) is a uniformly continuous function on
R, then the wavelet coefficient
bjk = 2j < f(x), ψ(2jx - k)
>
satisfies the bound |bjk| ≤ 2-1
ω(f,2-j-1), where ω is the modulus of continuity.
Assignment 7 - Due Wednesday, October 19.
- Read section 3.3, 3.4.
- Do the following problems.
- Problem 1, page 129 (§ 3.1).
- Problem 1, page 129 (§ 3.2).
- Problem 5, page 130 (§ 3.2).
- In the following, H is a complex Hilbert space and B(H) is the set
of bounded linear operators on H.
- Let L be in B(H). Show that ||L|| can be computed via either of the
formulas below.
- ||L|| = sup {||Lu|| : u ∈ H, ||u|| = 1}
- ||L|| = sup {|< Lu,v >| : u,v ∈ H, ||u|| = 1 and
||v|| = 1}
- Show that if P is the orthogonal projection onto a closed
subspace M, then ||P|| = 1. (Hint: use the Decomposition Theorem.)
- Show that using the operator norm on B(H) turns it into a normed,
linear space.
- Show that if M is a closed subspace of a Hilbert space H, then
(M⊥)⊥ = M. (In words, the orthogonal
complement of the orthogonal complement of M is M.)
Assignment 8 - Due Friday, November 4.
- Read sections 3.5, 3.6.
- Do the following problems.
- Problem 2, page 130 (§ 3.3).
- Problem 1(a), page 130 (§ 3.4).
- Problem 2(a), page 131 (§ 3.4).
- In the following, H is a Hilbert space and B(H) is the set of
bounded linear operators on H.
- Let L be in B(H). Show that ||L|| = sup {< L*Lu,
u>1/2 : u ∈ H,
||u|| = 1}.
- Let L be in B(H) and let M = sup {|< Lu, u>| : u ∈ H,
||u|| = 1}. Show that if H is a complex Hilbert space, then
M ≤ ||L|| ≤ 2M.
Hint: you will need to use the identity < L(u+v), u+v> - <
L(u-v), u-v> = 2< Lu,v>+2< Lv,u> first with v, and then
again with v replaced by i·v. (i2 = -1).
- For H = C2, let L =
0 1
0 0 .
Show that M = 1/2 and ||L|| = 1.
- For H = R2, which is a real Hilbert space, let L =
0 -1
1 0 .
Show that M = 0 and ||L|| = 1. Thus, (b) fails for real H.
- A sequence {fn} in H is said to be weakly
convergent to f∈H if and only if for every lim <
fn,g> = < f,g> for every
g∈H. When this happens, we write f = w-lim fn. 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.
- Let {φn} be any orthonormal sequence. Show that
φn converges weakly to 0. (Hint: use Bessel's
inequality.)
- Let K be a compact linear operator on a Hilbert space H. Show
that if fn weakly converges to f, then Kfn
actually converges to Kf.
Assignment 9 - Due monday, November 14.
- Read sections 4.1, 4.2
- Do the following problems.
- Problem 3, page 131 (§ 3.4).
- Problem 2, page 132 (§ 3.5).
- Problem 5, page 132 (§ 3.5).
- Problem 1(a,b), page 132 (§ 3.6).
- Problem 5, page 133 (§ 3.6).
Assignment 10 - Due Wednesday, November 23.
- Read sections 4.3, 4.4
- Do the following problems.
- Problem 2, page 177 (§ 4.1).
- Problem 4, page 178 (§ 4.1).
- Problem 5, page 178 (§ 4.1).
- Problem 9, page 178 (§ 4.1).
- Problem 3, page 179 (§ 4.2).
- Problem 9, page 179 (§ 4.2).
Assignment 11 - Due Monday, December 5.
- Read sections 4.5, 5.1
- Do the following problems.
- Problem 3, page 180 (§ 4.3) (Skip the extended operator
part.).
- Problem 6, page 180 (§ 4.3).
- Problem 6, page 181 (§ 4.5).
- Problem 9, page 181 (§ 4.5).
- Problem 5, page 207 (§ 5.1).
Updated 11/28/05 (fjn).