Math 641-600 — Fall 2010

Assignments

Assignment 1 - Due Wednesday, 9/8/2010.

Read sections 1.1-1.4.
Do the following problems.
1. Section 1.1: 5, 9(a). (In 9(a), do only the polynomials up to degree 2.)
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.)
  1. Show that there is a p for which
    min_{u ∈ U} ||v − u|| = ||v − p||,
    if and only if v − p is orthogonal to the subspace U.
  2. Show that p is unique, given that it exists for v.
  3. 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.)
3. In the previous problem, let U be finite dimensional, with a basis B = {u₁, ..., u_n}.
  1. Show that the n×n matrix G, with entries G_j,k = <u_k,u_j>, is invertible. (G is called a Gram matrix for the basis B.)
  2. Show that for every v ∈ V, the orthogonal projection of v onto U is given by
    p = ∑_k x_ku_k, where the x_j's satisfy the system <v,u_k > = ∑_j G_k,jx_j. That is, if we let d_k := <v,u_k >, then d = Gx.
  3. Show that if B is orthonormal, p = ∑_k <v,u_k >u_k.
  4. 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.
4. This problem concerns several important inequalities.
  1. Show that if α, β are positive and α + β =1, then for all u,v ≥ 0 we have
    u^αv^β ≤ αu + βv.
  2. Let x,y ∈ Rⁿ, and let p > 1 and define q by q^-1 = 1 - p^-1. Prove Hölder's inequality,
    |∑_j x_jy_j| ≤ ||x||_p ||y||_q.
    Hint: use the inequality in part (a), but with appropriate choices of the parameters. For example, u = (|x_j|/||x||_p)^p
  3. 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.
1. Section 1.3: 2, 5.
2. Section 1.4: 3, 4.
3. 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.
4. Let U be a unitary, n×n matrix. Do the following.
  1. Show that < Ux, Uy > = < x, y >.
  2. Show that the eigenvalues of U all lie on the unit circle, |λ|=1.
  3. 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.)
  4. 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.)
5. Consider a spring system with 3 equal masses (m_j = m), and 3 equal springs (k_j = k) connecting them, arranged on a circle. Here, the displacement u_j is an angle θ_j, 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.
1. Section 2.1: 3, 5, 10
2. Show that ℓ² is a Hilbert space -- i.e., that it's complete -- under the inner product ⟨x,y⟩ = ∑_j x_jy_j. (j = 0,...)
3. Fix f ∈ C[0,1] and let ω(f;δ) be the modulus of continuity of f.
  1. Fix δ. Let S_δ = { 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 S_δ
  2. 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.)
  3. Show that lim_δ↓0 ω(f;δ) = 0.
4. Let f ∈ C⁽²⁾[0,1]. Show that if f_n is the equally-spaced interpolating spline for f, then
  ||f − f_n||_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].
5. 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.
1. Section 2.1: 9
2. Section 2.2: 7, 8
3. 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
  lim_{s ↓ 0} F(s) = F(0)= ∫₀^∞f(t)dt
4. 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.
5. 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.
6. Compute the Fourier series for the following functions. For each of these, write out the corresponding version of Parseval's identity.
  1. f(x) = x, 0≤ x ≤ 2π (π = "pi".)
  2. f(x) = |x|, − π ≤ x ≤ π
  3. f(x) = e^2x, 0≤ x ≤ 2π (complex form).
  4. 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 ∈ S^1/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)|_a^b. (Note: p′′′ is constant on [a,b].) Applying these appropriately will solve the problem.
Do the following problems.
1. Section 2.2: 25, 26(a), 27(a)
2. Consider a 2π periodic function f with Fourier series f(t) = ∑_n c_ne^int. Show that if f is C^(k), then |c_n| ≤ C |n|^{− k} for all n ≠ 0.
3. Prove the Convolution Theorem for the DFT. (See Notes on the Discrete Fourier Transform, pg. 3.)
4. Let a, x, y be column vectors with entries a₀, ..., a_n-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.
  1. 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
```
  2. Such matrices are called cyclic. Use the DFT and the convolution theorem to find the eigenvalues of a cyclic matrix A.
5. Variational/Finite-element problem. We want to solve the boundary value problem (BVP): −u'' = f(x), u(0) = u(1) = 0.
  1. 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.
  2. Let V = S^h(1,0), with h = 1/n. Thus V is spanned by φ_j(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 α_jφ_j(x) ∈ V, where the α_j's satisfy Φα = β, with
    β_j = ⟨ y,φ_j ⟩_H = ∫₀¹ f(x) φ_j(x) dx, j=1 ... n-1 and Φ_kj = ⟨ φ_j, φ_k ⟩_H.
  3. Show that Φ_kj = ⟨ φ_j, φ_k ⟩_H is given by
    Φ_j,j = 2n, j = 1 ... n-1
    Φ_j,j-1 = - n, j = 2 ... n-1
    Φ_j,j+1 = - n, j = 1 ... n-2
    Φ_j,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.
1. 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.)
2. Let M be a subspace of a Hilbert space H. Show that M is closed if and only if M = (M^⊥)^⊥.
3. 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.
4. 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.
  1. P is a bounded linear operator, with ||P||=1.
  2. Range(P) = M and Null(P) = M^⊥.
  3. P is self-adjoint, i.e. P^* = P.
5. 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.
  1. Show that, in the inner product above, H is a Hilbert space.
  2. 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.
1. Section 3.2: 3(b,d)
2. Let L be a bounded linear operator on Hilbert space H. Show that the two formulas for ||L|| are equivalent:
  1. ||L|| = sup {||Lu|| : u ∈ H, ||u|| = 1}
  2. ||L|| = sup {|< Lu,v >| : u,v ∈ H, ||u|| = 1 and ||v|| = 1}
3. 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.
  1. 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.
  2. 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?
4. (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 u_M to the problem for M,
  < u_M, 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 v_M ∈ M satisfies
  ||u − v_M||_H¹ = min{|| u− v||_H¹ ∀ v ∈ M},
  then there is a constant C such that
  C ||u − u_M||_H¹ ≤ ||u − v_M||_H¹ ≤ ||u − u_M||_H¹.
5. A sequence {f_n} in a Hilbert space H is said to be weakly convergent to f∈H if and only if lim_{n
  → ∞} < f_n,g> = < f,g> for every g∈H. When this happens, we write f = w-lim f_n. For example, in class we showed that if {φ_n} is any orthonormal sequence, then φ_n 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 ||f_n|| ≤ C for all n. Prove the following:
  Let K be a compact linear operator on a Hilbert space H. Show that if f_n weakly converges to f, then Kf_n converges strongly to Kf — that is, lim_{n →
  ∞} || Kf_n - Kf || =0.
  Hint: Suppose this doesn't happen, then there will be a subsequence of {f_n}, say {g_k}, such that
  || Kg_k - 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.
1. Section 3.4: 1(a,b), 2
2. Section 3.5: 1(b), 2(b)
3. Section 3.6: 5
4. Section 4.1: 4
5. Let K be a compact, self-adjoint operator. Show that the only possible limit point of the set of eigenvalues {μ_j} of K is 0 — i.e., μ ≠0 is never a limit point of the set of eigenvalues.

Updated 11/18/2010.