Math 641-600 — Fall 2014

• Read sections 1.1-1.4
• Do the following problems.
  1. Section 1.1: 3(c), 5, 7(a), 8
  2. Section 1.4: 3
  3. Let $U$ be a subspace of an inner product space $V$, with the inner product and norm being $\langle\cdot,\cdot \rangle$ and $\|\cdot\|$. Also, let $v$ be in $V$. (Do not assume that $U$ is finite dimensional or use arguments requiring a basis.)
     1. Fix $v\in V$. Show that if there exists $p\in U$ such that $p$ satisfies either (a) $\min_{u\in U}\|v-u\| = \|v-p\|$ or (b) $v-p\in U^\perp$, then it satisfies both (a) and (b). Moreover, $p$ is unique. (If $v\in U$, then $p=v$.)
     2. Suppose $p$ exists for every $v\in V$. Since $p$ is uniquely determined by $v$, we may define a map $P: V \to U$ via $Pv:=p$. Show that $P$ is a linear map and that $P$ satisfies $P^2 = P$. ($P$ is called an orthogonal projection. The vector $p$ is the orthogonal projection of $v$ onto $U$.)
     3. If the projection $P$ exists, show that for all $w,z\in V$, $\langle Pw,z\rangle = \langle Pw,Pz\rangle= \langle w,Pz\rangle$. Use this to show that $U^\perp= \{w\in V\colon Pw=0\}$.
     4. Suppose that the projection $P$ exists. Show that $V=U\oplus U^\perp$, where $\oplus$ indicates the direct sum of the two spaces.

  4. Let $U$ and $V$ be as in the previous exercise. Suppose that $U$ is finite dimensional and that $B=\{u_1,u_2,\ldots,u_n\}$ is an ordered basis for $U$. In addition, let $G$ be the $n\times n$ matrix with entries $G_{jk}= \langle u_k,u_j\rangle$.
     1. Let $v\in V$ and $d_k =: \langle v,u_k\rangle$. Show that $p$ exists for every $v$ and is given by $p=\sum_j x_j u_j\in U$, where the $x_j$'s satisfy the normal equations, $d_k = \sum_j G_{kj}x_j$. (Hint: use exercise 2 above.)
     2. Explain why the projection $P$ exists. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

Assignment 2 - Due Wednesday, September 17.

• Read sections 2.1 and 2.2
• Do the following problems.
  1. Section 1.2: 9, 10(a,b)
  2. Section 1.3: 2(b)
  3. Section 1.4: 4
  4. Let V be an n dimensional vector space and suppose L:V→V is linear.
     1. Let E={φ₁,.., φ₁} be a basis for V and let A be the matrix of L relative to E. Show that the characteristic polynomial p_A(λ) := det(A − λ I) is independent of the choice of E, and so p_L(λ) := p_A(λ) is well defined, in the sense that it is independent of the choice of basis for V.
     2. Use the previous part to show that Trace(L) := Trace(A) and det(L) := det(A) are also well defined,
     3. Let $L:P_2\to P_2$ be given by $L(p)= \big((1-x^2)p'\big)' + 7p$. Find Trace(L) and det(L).

  5. (This is a generalization of Keener's problem 1.3.5.) Let $A$ be a self-adjoint matrix with eigenvalues $\lambda_1\ge \lambda_2,\ldots,\ge \lambda_n$. Show that for $ 2\le k < n$ we have \[ \max_U \sum_{j=1}^k \langle Au_j,u_j \rangle =\sum_{j=1}^k \lambda_j, \] where $U=\{u_1,\ldots,u_k\}$ is any o.n. set. (Hint: Put $A$ in diagonal form and use a judicious choice of $B$.)

  6. Let U be a unitary, n×n matrix. Show that the following hold.
     1. < Ux, Uy > = < x, y >
     2. The eigenvalues of U all lie on the unit circle, |λ|=1.
     3. Show that U is diagonalizable. (Hint: follow the proof for the self-adjoint case.)

  7. Let $k(x,y) = x+ 3x^2y + xy^2$ and $\langle f,g\rangle=\int_{-1}^1 f(x)g(x)(1+x^2)dx$. Consider the operator $Lu=\int_{-1}^1 k(x,y) u(y)dy$. In the notes, we have shown that $L:P_2\to P_2$.
     1. Relative to the inner product above, find $L^\ast$ and $\text{Null}(L^\ast)$.
     2. Find a condition on $q\in P_2$ for which $Lp=q$ always has a solution. Is this different from what was in the notes?

Assignment 3 - Due Wednesday, September 24.

• Read sections 2.1 and 2.2
• Do the following problems.
  1. Section 2.1: 3, 5, 6
  2. Show that $\ell^2$, under the inner product $\langle x,y\rangle = \sum_{j=1}^\infty x_j \overline{y_j}$, is a Hilbert space.
  3. 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.
  4. Let f(x) : x², -1 ≤ x ≤ 2. Find f ⁻¹(Ej) for E_j = [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

Assignment 4 - Due Wednesday, October 1.

• Read sections 2.1.1 and 2.2.
• Do the following problems.
  1. Section 2.1: 10, 11
  2. Section 2.2: 1 (Do $w=1$ and $w=\frac{1}{\sqrt{1-x^2}}$.)
  3. Let $F(s) := \int_0^\infty e^{-st}f(t)dt $ be the Laplace transform of $f \in L^1([0,\infty))$. Use the Lebesgue dominated convergence theorem to show that $F(s)$ is continuous from the right at $s=0$.
  4. Let $\{f_n\in C^1[0,1]\}$. Note that $\{f_n\}$ also belongs to $H^1[0,1]$. Show that if $f_n$ is Cauchy in $H^1[0,1]$, then, in the $C[0,1]$ norm, $\{f_n\}$ actually converges to a continuous function $f$.

  5. Let $U:=\{u_j\}_{j=1}^\infty$ be an orthonormal set in a Hilbert space $\mathcal H$. Show that the two statements are equivalent. (You may use what we have proved for o.n. sets in general; for example, Bessel's inequality, minimization properties, etc.)
     1. $U$ is maximal in the sense that there is no non-zero vector in $\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a proper subset of any other o.n. set in $\mathcal H$.)
     2. Every vector in $\mathcal H$ may be uniquely represented as the series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

• Read sections 2.2.2-2.2.4
• Do the following problems.
  1. Section 2.2: 8(a,b,d), 9

  2. Let $0\le \delta \le 1$. We define the modulus of continuity for $f\in C[0,1]$ by $ \omega(f;\delta) := \sup_{\,|\,s-t\,|\,\le\, \delta,\,s,t\in [0,1]}|f(s)-f(t)|. $
     1. Explain why $\omega(f;\delta)$ exists for every $f\in C[0,1]$.
     2. 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, S_δ is in the set of all ε such that |f(t) − f(s)| < ε holds for all |s − t| ≤ δ. Show that
        ω(f;δ) = inf S_δ
     3. 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.)
     4. Show that lim _δ↓0 ω(f;δ) = 0.
  3. Calculus problem: Let g be C² on an interval [a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
     ||g||_C[a,b] ≤ (h²/8) ||g′′||_C[a,b].
     Give an example that shows that 1/8 is the best possible constant.

  4. Use the previous problem to show that if f ∈ C²[0,1], then the equally spaced linear spline interpolant f_n satisfies
     ||f − f_n||_C[0,1] ≤ (8n²) ^{− 1} ||f′′||_C[0,1]

• Read sections 2.2.7
• Do the following problems.
  1. Section 2.2: 14

  2. Prove this: Let $g$ be a $2\pi$-periodic piecewise continuous function. Then, $\int_{-\pi+c}^{\pi+c} g(u)du$ is independent of $c$. (Remark: This holds for $g$ integrable on each bounded interval of $\mathbb R$.)

  3. Compute the Fourier series for the following functions. For each of these, write out the corresponding version of Parseval's identity. (This is just so you see the form of the identity, nothing more.) In each case, estimate the L² error made when the series is truncated at a large integer N.
     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. Consider the series ∑_n c_n e^inx, where ∑_n |c_n| < ∞. Show that ∑_n c_n e^inx converges uniformly to a continuous function f(x) and that the series is the Fourier series for f.

  5. Prove the Convolution Theorem for the DFT. (See Notes on the Discrete Fourier Transform, pg. 3.)

Assignment 7 - Due Wednesday, November 5.

• Read sections 3.1 and 3.2.
• Do the following problems.
  1. Section 2.2: 25(a,b), 26(b), 27(a)

  2. We want to use the Galerkin method to numerically solve the boundary value problem (BVP):  −u" = f(x), u(0) = u(1) = 0, f ∈ C[0,1]
     1. Weak form of the problem. Let H₀ be the set of all continuous functions vanishing at x = 0 and x = 1, and having L² derivatives. Also, let H₀ have the inner product:
        ⟨u,v⟩_H0 = ∫₀¹ u ′(x) v ′(x) dx.
        Use integration by parts to convert the BVP into its ``weak'' form:
        ⟨u,v⟩_H0 = ∫₀¹ f(x) v(x) dx for all v ∈ H₀.

     2. Consider S₀ := {s ∈S^1/n(1,0) : s(0)=s(1)=0}. Show that S₀ is spanned by φ_j(x) := N₂(nx-j+1), j = 1 ... n-1. (Here, N₂(x) is the linear B-spline.)

     3. Show that the least-squares approximation s ∈ S₀ to the solution u is given by s = ∑_j α_jφ_j(x), where the α_j's satisfy Gα = β, with
        β_j = ⟨ y,φ_j ⟩_H0 = ∫₀¹ f(x) φ_j(x) dx, j=1 ... n-1 and G_kj = ⟨ φ_j, φ_k ⟩_H0.

     4. Show that G_kj = ⟨ φ_j, φ_k ⟩_H0 is given by
        G_j,j = 2n, j = 1 ... n-1
        G_j,j-1 = - n, j = 2 ... n-1
        G_j,j+1 = - n, j = 1 ... n-2
        G_j,k = 0, all other possible k.
        

Assignment 8 - Due Wednesday, November 12.

• Read sections 3.3 and 3.4.
• Do the following problems.
  1. Section 3.2: 3(d) (Assume the appropriate operators are closed and that λ is real.)

  2. Section 3.3: 2 (Assume the appropriate operators are closed and that λ is real.)

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

  4. (DFT problem.) Let α, ξ, η be n-periodic sequences, and let a, x, y be column vectors with entries a₀, ..., a_n-1, etc. 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. Such matrices are called cyclic. Use the DFT and the convolution theorem to find the eigenvalues of A. An example of a cyclic matrix is given below. \[ \begin{pmatrix} 3 &5 &4 &1 \\ 1 &3 &5 &4 \\ 4 &1 &3 &5\\ 5 &4 &1 &3 \end{pmatrix} \]

  5. Let L be a bounded linear operator on Hilbert space $\mathcal H$. Show that these two formulas for $\|L\|$ are equivalent:
     1. $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
     2. $\|L\| = \sup \{|\langle Lu,v\rangle| : u,v \in {\mathcal H},\ \|u\|=\|v\|=1\}$

  6. Consider the boundary value problem $-u''(x)=f(x)$, where $0\le x \le 1$, $\, f\in C[0,1]$, $\, u(0)=0$ and $u'(1)=0$.
     1. Verify that the solution is given by $u(x) = \int_0^1 k(x,y)f(y)dy$, where \[ k(x,y) = \left\{ \begin{array}{cl} y, & 0 \le y \le x, \\ x, & x \le y \le 1. \end{array} \right. \]
     2. Let $L$ be the integral operator $L\,f = \int_0^1 k(x,y)f(y)dy$. Show that $L:C[0,1]\to C[0,1]$ is bounded and that the norm $\|L\|_{C[0,1]\to C[0,1]}\le 1$. Actually, $\|L\|_{C[0,1]\to C[0,1]}=1/2$. Can you show this?

     3. Show that $k(x,y)$ is a Hilbert-Schmidt kernel and that $\|L\|_{L^2\to L^2} \le \sqrt{\frac{1}{6}}$.

  7. Finish the proof of the Projection Theorem: If for every $f\in \mathcal H$ there is a $p\in V$ such that $\|p-f\|=\min_{v\in V}\|v-f\|$ then $V$ is closed.

Assignment 9 - Due Wednesday, November 19.

• Read sections 3.5 and 3.6.
• Do the following problems.
  1. Section 3.4: 2(b)

  2. Consider space H = {v ∈ H¹[0,1] : v(0) = v(1) = 0}. Let q(x) be continuous and strictly positive on [0,1]. On H, define the (real) inner product
     < f,g >_H = ∫₀¹(f ′(x) g ′(x) +q(x)f(x)g(x))dx,
     A weak solution u to the boundary value problem (BVP)
     -u''+q(x)u = f(x), u(0) = u(1) = 0, f ∈ L²[0,1].
     is a function u ∈ H such that for all v ∈ H we have < u, v >_H = ∫₀¹v(x)f(x)dx.
     1. Show that, in the inner product above, H is a Hilbert space. (Hint: use the fact that H¹ is a Hilbert space.)
     2. Show that the BVP has a unique weak solution in H.

  3. 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, if {φ_n} is any orthonormal sequence, then φ_n converges weakly to 0. Given that every weakly convergent sequence is a bounded sequence (i.e. 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. If f_n converges weakly 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 {f_nk}, such that
     || Kf_nk - Kf || ≥ ε
     for all k. Use this and the compactness of K to arrive at a contradiction.

  4. Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty \in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,|a_j|^2\le 1 \}$. Show that $S$ is a compact subset of $\ell^{\,2}$.
  5. Let L be a bounded self-adjoint linear operator on a Hilbert space $\mathcal H$. Show that these two formulas for $\|L\|$ are equivalent:
     1. $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
     2. $\|L\| = \sup \{|\langle Lu,u\rangle| : u\in {\mathcal H},\ \|u\|=1\}$

Assignment 10 - Due Wednesday, December 3.

• Read sections 3.5, 3.6, and 4.1.
• Do the following problems.
  1. Section 3.4: 2(c,d), 3
  2. Section 3.5: 1(b), 2(a)
  3. Section 3.6: 1(a), 5

  4. Let K be a compact, self-adjoint operator and let M be the span of the set of eigenvectors {φ_j} corresponding to all eigenvalues λ_j ≠ 0. (Note: both M and M^⊥ may be infinite dimensional.)
     1. Show that M and M^⊥ are both invariant under K.
     2. Show that K restricted to M^⊥ is compact.
     3. Show that either M^⊥ = {0} or that it is the eigenspace for λ = 0.
     4. Show that one may choose a complete, orthonormal set from among the eigenvectors of K.

  5. Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) := \int_0^1(2+st+u(s)^2)^{-1}ds$, $0\le t\le 1$. Let $\| \cdot \|:=\|\cdot \|_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,|\, \|u\|\le r\}$.
     1. Show that $F: B_1\to B_{1/2}\subset B_1$.
     2. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha<1$ -- i.e., $\|F[u]-F[v]\|\le \alpha \|u-v\|$.
     3. Show that $F$ has a fixed point in $B_1$.

Extra Problems - These are not to be handed in.

• Section 4.1: 1(b), 4, 6
• Section 4.2: 1, 4, 8
• Section 4.3: 3
• Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
  1. Show that the Green's function for this problem is \[ G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right. \]
  2. Verify that $0$ is not an eigenvalue for $Kf(x) := \int_0^1G(x,y)f(y)dy$.
  3. Verify that $u=x^2$ satisfies the boundary conditions required for it to be in the domain of $L$ and that $\langle Lu,u \rangle < 0$. Also verify if we let $u=x(x-1)^2$, then $u$ is in the domain for $L$ and that $\langle Lu,u \rangle > 0$. Thus $L$ does not satisfy the conditions of Keener's Theorem 4.7.
  4. Even though the Theorem 4.7 is not satisfied, the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$, Explain.

Updated 12/8/2014.