Math 641-600 Assignments

Assignment 1 Due Friday, 9/11/09.

• Read sections 1.1-1.4.
• Do the following problems.
  1. Section 1.1: 4, 5, 7, 9(c). (In 9(c), do only the first 3 polynomials.)
  2. 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 there is a u₀ for which

     min _u ||v-u|| = ||v-u₀||,

     if and only if v-u₀ is orthogonal to every u in U. Also, show that u₀ is unique, if it exists.
  3. Section 1.2: 6(a,c), 8, 9.
  4. Section 1.3: 2.

Assignment 2 - Due Wednesday, 9/23/09.

• Read sections 2.1-2.2.2.
• Do the following problems.
  1. Section 1.3: 3, 4, 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^⊥. (You may need to look up the definition of ⊕, which symbolizes the direct sum of vector spaces.) 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. (Scalars must be complex).
     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? decreases?

Assignment 3 - Due Wednesday, 9/30/09.

• Read sections 2.1-2.2.2.
• Do the following problems.
  1. 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,...)
  2. Fix f∈ C[0,1] and let ω(f;δ) be the modulus of continuity of f.
     1. Show that ω(f;δ) = inf{ ε > 0 | for all t,s in [0,1] satisfying |t − s| ≤ δ one has |f(t) − f(s)| < ε}.
     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. In class, we said that lim _δ↓0 ω(f;δ) = 0. Show that this is true.
  3. 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}.
  4. Assume that if A ⊂ [0,1], then m_∗(A) ≤ m^∗(A). Use this to show that if B ⊂ A ⊂ [0,1] and if m(A) = 0, then B is measurable and m(B) = 0.

Assignment 4 - Due Friday, October 9, 2009.

• Read sections 2.2.1-2.2.3.
• Do the following problems.
  1. Let f_n(x) = n^3/2 x e^{-n x}, where x ∈ [0,1] and n = 1, 2, 3, ....
     1. Verify that the pointwise limit of f_n(x) is f(x) = 0.
     2. Show that ||f_n||_C[0,1] → ∞ as n → ∞, so that f_n does not converge uniformly to 0.
     3. Find a constant C such that for all n and x fixed f_n(x) ≤ C x^-1/2, x ∈ (0,1].
     4. Use the Lebesgue dominated convergence theorem to show that
        lim_{n→ ∞} ∫₀¹ f_n(x)dx = 0.
  2. Find the best least-squares approximation in L²[0,1] to e^x from span{1,x}. Make a plot of e^x and the line that best fits e^x.
  3. Section 2.2: 8.
  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. 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. Show that
        ⟨f,g⟩_H = ∫₀¹ f ′(x) g ′(x) dx,
        is an inner product for H. (Hint: To show positivity, use the previous problem.)
     2. Let f(x) = x² in the rest of the problem. Find the exact solution to the BVP for this choice of f.
     3. Let φ_j(x) := N₂(nx-j+1), where N₂(x) is the linear B-spline, or ``tent'' function, defined on pg. 81. Find the β_j's, where
        β_j = ⟨ y,φ_j ⟩_H = ∫₀¹ f(x) φ_j(x) dx, j=1 ... n-1.

     4. Show that Φ_kj = ⟨ φ_j, φ_k ⟩_H, the k-j entry in the Gram matrix for the problem, satisfies
        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.
        For example, if n=5, then Φ is

        10  -5   0   0
        -5  10  -5   0
         0  -5  10  -5 
         0   0  -5  10
        

     5. 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 α_j φ_j(x),
        which is also the least squares approximation to y in the inner product ⟨ ·,· ⟩_H defined above.
  6. Bonus.   Section 2.2: 4.

Assignment 5 - Due Monday, November 9, 2009.

• Read sections 2.2.6, 2.2.7, 3.1.
• Do the following problems.
  1. 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
     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.
  2. Let f_N be the partial sum for the series in (b) above. Estimate the error ||f - f_N|| when the norm is for L² and when the norm C[− π,&pi:]. (Hint: use the esitmate form the integral test.)
  3. 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.
  4. Prove the Convolution Theorem for the DFT. (See Notes on the Discrete Fourier Transform, pg. 3.)
  5. 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, repectively.
     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.
  6. Let ψ(x) be the Haar wavelet. Show that if f ∈ L²(R) is a uniformly continuous function on R, then the wavelet coefficient
     d^j_k = 2^j < f(x), ψ(2^jx - k) >
     satisfies the bound | d^j_k | ≤ 2^-1 ω(f,2^-j-1), where ω is the modulus of continuity.
  7. 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 6 - Due Monday, December 2, 2009.

• Do the following problems.
  1. Section 2.2: 26. (The definition of projection applies to to Banach spaces as well as Hilbert spaces. The two parts of the problem involve Banach spaces.)

  2. Prove Proposition 0.1 in Notes on Daubechies' Wavelets

  3. Prove the polarization identity for u,v in a Hilbert space H:
     ||u+v||² + ||u-v||² = 2(||u||² + ||v||²).
     

  4. Let M be a subspace of a Hilbert space H. Show that M is closed if and only if M = (M^&perp)^&perp.

  5. Let V be a Banach space. Show that a linear operator L:V → V is bounded if and only if L is continuous.

  6. 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. Define the operator P:H → H by Ph = p. The operator P is called the projection of H onto M. Show that the following are true.
     1. P is a bounded linear operator, with ||P||=1.
     2. P² = P.
     3. Range(P) = M and Null(P) = M^⊥.
     4. P is self-adjoint, i.e. P^* = P.
  7. Consider space H = comprising all functions f in Sobolev space H¹[0,1] that satsify f(0) = f(1) = 0. Let v(v) 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.

  8. Section 3.2 problem 3(d), page 128. (Assume the appropriate operators are closed and that λ is real.)

  9. Section 3.3 problem 2, page 129. (Assume the appropriate operators are closed and that λ is real.)

Updated 11/20/09 (fjn).