# Math 641-600 — Fall 2018

## Assignments

Assignment 1 - Due Wednesday, September 5, 2018

• Do the following problems.
1. Section 1.1: 4, 5, 7(a), 8, 9(a) (Do the first 3, but without software.)
2. Section 1.2: 9
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 there is a unique vector $p \in U$ that satisfies $\min_{u\in U}\|v-u\| = \|v-p\|$ if and only if $v-p\in U^\perp$.
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. (This is easy, but important.)

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. Show that $G$ is positive definite and thus invertible.
2. 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, $d_k = \sum_{j=1}^n G_{kj}x_j$.
3. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

Assignment 2 - Due Wednesday, September 12, 2018.

• Read the notes on Banach spaces and Hilbert Spaces, and sections 2.1 and 2.2 in Keener.

• Do the following problems.
1. Section 1.2: 10(a,b) Hint for !0(a): You may choose the norms $\| \phi_j\|$ and $\|\psi_k\|$ to be any (convenient) positive numbers.

2. Section 1.3: 2, 3
3. Find the set of biorthogonal vectors corresponding to the set $\left\{\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix}2 \\ 1\\ 0\end{pmatrix}\right\}$.  Suppose that $\{\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_n\}$ is a set of linearly independent vectors in $\mathbb R^n$. What is the corresponding set of biorthogonal vectors?

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
uαvβ ≤ 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 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
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.

5. Find the $QR$ factorization for the matrix $A=\begin{pmatrix} 1 & 2 & 0\\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$. Use it to solve $Ax=b$, where $b=\begin{pmatrix} 1\\ 3\\ 7 \end{pmatrix}$.

6. Let $\mathbf y\in \mathbb R^n$. Use the normal equations for a minimization problem to show that the minimizer of $\| \mathbf y - A\mathbf x\|$ is given by $\mathbf x_{min} = R^{-1}Q^\ast \mathbf y$. ($Q^\ast=Q^T$, since we are dealing with real scalars.)

7. 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. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Assignment 3 - Due Wednesday, September 19, 2018.

• Read Keener's sections 2.1 and the notes on Lebesgue integration.
• Do the following problems.

1. Section 2.1: 5

2. Before one can define a norm or inner product on some set, one has to show that the set is a vector space -- i.e., that linear combinations of vectors are in the space. Do this for the spaces of sequences below. The inequalities from the previous assignment will be useful.
1. $\ell^2=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^2\}$
2. $\ell^p=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^p\}$, all $1\le p<\infty$, $p\ne 2$.
3. $\ell^\infty = \{x=\{x_n\}_{n=1}^\infty\colon \sup_{1\lr j}|x_j|<\infty \}$.

3. Show that, for all $1\le p <\infty$, $\|x\|_p := \big(\sum_{j=1}^\infty |x_j|^p \big)^{1/p}$ defines a norm on $\ell^p$.

4. Show that $\ell^2$ is an inner product space, with $\langle x,y\rangle = \sum_{j-1}^\infty x_j \bar y_j$ being the inner product, and that with this inner product it is a Hilbert space. Bonus: show that it is separable.

5. Let $C^1[0,1]$ be the set of all continuously differentiable real-valued functions on $[0,1]$. Show that $C^1[0,1]$ is a Banach space if $\|f\|_{C^1} := \max_{x\in [0,1]}|f(x)| + \max_{x\in [0,1]}|f'(x)|$.

6. Let $f\in C^1[0,1]$. Show that $\|f\|_{C[0,1]}\le C\|f\|_{H^1[0,1]}$, where $C$ is a constant independent of $f$ and $\|f\|_{H^1[0,1]}^2 := \int_0^1\big( |f(x)|^2 + |f'(x)|^2\big)dx$.
7. A measurable function whose range consists of a finite number of values is a simple function — see Lebesgue integration, p. 5. Use the definition of the Lebesgue integral in in terms of Lebesgue sums, from eqn. 2, to show that, in terms of this definition, the integral of a simple function ends up being the one in eqn. 3 on p. 6.

Assignment 4 - Due Wednesday, September 26, 2018.

Assignment 5 - Due Wednesday, October 3, 2018.

Assignment 6 - Due Wednesday, October 10, 2018.

• Read sections 2.2.2-2.2.4, the notes on Fourier series, and the notes on the discrete Fourier transform.
• Do the following problems.
1. Section 2.2: 14

2. Prove this: Let $g$ be a $2\pi$ periodic function (a.e.) that is integrable on each bounded interval in $\mathbb R$. Then, $\int_{-\pi+c}^{\pi+c} g(u)du$ is independent of $c$. In particular, $\int_{-\pi+c}^{\pi+c} g(u)du=\int_{-\pi}^\pi g(u)du$.

3. Compute the Fourier series for the following functions.
1. f(x) = x,   0≤ x ≤ 2π
2. f(x) = |x|,   − π ≤ x ≤ π
3. f(x) = e2x,  − π ≤ x ≤ π (complex form).

4. Compute the complex form of the Fourier series for $f(x) = e^{2x}$, $0 \le x \le 2\pi$. Why is this different from 3(c) above? Use this Fourier series and Parseval's theorem to sum the series $\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.

5. The following problem is aimed at showing that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.
1. Consider the series ∑n cn einx, where ∑n |cn| < ∞. Show that ∑n cn einx converges uniformly to a continuous function f(x) and that the series is the Fourier series for f. (It's possible for a trigonometric series to converge pointwise to a function, but not be the Fourier series for that function.)

2. Use the previous problem to show that if $f$ is a continuous, piecewise smooth $2\pi$-periodic function, then the FS for $f$ converges uniformly to $f$. (Hint: Show that if $f'\in L^2[-\pi,\pi]$, then series $\sum_{k=-\infty}^\infty k^2|c_k|^2$ is convergent.)

3. Apply this result to show that the FS for a linear spline $s(x)$, which satisfies $s(-\pi)=s(\pi)$, is uniformly convergent to $s(x)$. Show that such splines are dense in $L^2[-\pi,\pi]$.

4. Show that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.

6. Let $\mathcal S_n$ be the set of $n$-periodic, complex-valued sequences.

1. Suppose that $\mathbf x \in \mathcal S_n$. Show that $\sum_{j=m}^{m+n-1}{\mathbf x}_j = \sum_{j=0}^{n-1}{\mathbf x}_j$. (This is the DFT analogue of problem 1 above.)

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

Assignment 7 - Due Wednesday, October 24, 2018.

• Read section 2.2.7 and the notes on Splines and Finite Element Spaces.
• Do the following problems.
1. Section 2.2: 18(a,d). (Both of these use the formula $N_m(x)=\frac{x}{m-1}N_{m-1}(x)+\frac{m-x}{m-1}N_{m-1}(x-1)$, together with induction).

2. Let $f(t)=10\cos(2t)$ and consider the ODE $u''+2u'+2u=f(t)$.

1. Verify that the general solution to the equation is $u=Ae^{-t}\cos(t)+ Be^{-t}\sin(t) +2\sin(2t)-\cos(2t)$; consequently the "steady state" periodic solution is $u_p(t)=2\sin(2t)-\cos(t)$.

2. Let $n=2^L$ and $h=\frac{2\pi}{n}$. For $L=3,5,8,\text{and}\ 10$, sample $f$ at $jh$, $j=0\ldots n-1$; let $f_j:=f(jh)$. Use your favorite program to find the FFT of $\{f_0,f_1,\ldots,f_{n-1}\}$ and, using the method outlined in the notes on the discrete Fourier transform, find $\hat u_k$. Finally, apply your program's inverse FFT to the $\hat u_k$'s to obtain the approximation $u_j$ to $u_p$ at $jh$. For each $L$, plot the $u_j$'s and the $u_p(jh)$'s. The $u_j$'s may have a small complex part due to roundoff error; just plot the real parts of the $u_j$'s you found by the procedure above. Be sure to label your plots.

3. For each $L$ plot the error $\{ |u_0-u_p(0)|,|u_1-u_p(h)|,\ldots, |u_{n-1}-u_p((n-1)h)|\}$; again, label your plots.

3. Let α, ξ, η be n-periodic sequences, and let a, x, y be column vectors with entries a0, ..., an-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 $\mathbf a$, and whose remaining columns are $\mathbf a$ with the entries cyclically permuted. Such matrices are called cyclic. Use the DFT and the convolution theorem to find the eigenvalues of the a cyclic matrix. Use this method, along with your favorite software, to find the eigenvalues and eigenvectors of the matrix below. (For this matrix, $\mathbf a =(3\ 1\ 4\ 5)^T$.) $\begin{pmatrix} 3 &5 &4 &1 \\ 1 &3 &5 &4 \\ 4 &1 &3 &5\\ 5 &4 &1 &3 \end{pmatrix}$

4. Let $S^{1/n}(1,0)$ be the space of piecewise linear splines, with knots at $x_j=j/n$, and let $N_2(x)$ be the linear B-spline ("tent function", see Keener, p. 81 or my notes on splines.)

Assignment 8 - Due Wednesday, October 31, 2018.

Assignment 9 - Due Wednesday, November 7, 2018.

Assignment 10 - Due Wednesday, November 14, 2018.

• Read sections 3.3-3.5, and my notes on Compact Operators, and on the Closed Range Theorem.
• Do the following problems.

1. Section 3.3: 1 (Assume the appropriate operators are closed and that λ is real.)

2. Section 3.4: 2(b)

3. Consider the Hilbert space $\mathcal H=\ell^2$ and let $S=\{x=(x_{1}\ x_{2}\ x_3\ \ldots)\in \ell^2: \sum_{n=1}^\infty (n^2+1)|x_n|^2 <1\}$. Show that $S$ is a precompact subset of $\ell^2$.

4. Let $S$ be a bounded subset (not a subspace!) of a Hilbert space $\mathcal H$. Show that $S$ is precompact if and only if every sequence in $S$ has a convergent subsequence. (Note: If $S$ is just precompact, the limit point of the sequence may not be in $S$, because $S$ may not be closed.)

5. Show that every compact operator on a Hilbert space is bounded.

6. Consider the finite rank (degenerate) kernel
k(x,y) = φ1(x)ψ1(y) + φ2(x)ψ2(y), where φ1 = 6x-3, φ2 = 3x2, ψ1 = 1, ψ2 = 8x − 6.
Let Ku= ∫01 k(x,y)u(y)dy. Assume that L = I-λ K has closed range,
1. For what values of λ does the integral equation
u(x) - λ∫01 k(x,y)u(y)dy =f(x)
have a solution for all f ∈ L2[0,1]?
2. For these values, find the solution u = (I − λK)−1f — i.e., find the resolvent.
3. 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?

7. 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) and let N:= sup {|< Lu, u>| : u ∈ H, ||u|| = 1}.

Assignment 11 - Due Monday, November 26, 2018.

• Read sections 3.3-3.5, and my notes on and my notes on Spectral Theory for Compact Operators.
• Do the following problems.

1. Section 3.4: 2(a)

2. Let $L\in \mathcal B(\mathcal H)$. Suppose that for all $f\in N(L)^\perp$ there is a constant $c>0$ such that $\|Lf\|\ge c\|f\|$, where $c$ is independent of $f$. Show that $R(L)$ is closed.

3. Finish the proof of Proposition 2.5 in my notes on Compact Operators

4. Consider the kernel $k(x,y)=\min(x,y)$, $0\le x,y\le 1$.
1. Show that $Ku=\int_0^1 k(x,y)u(y)dy$ is a compact, self-adjoint operator operator.
2. Let $U(x)=\int_0^x u(y)dy - \int_0^1 u(y)dy$. Show that $Ku(x) = -\int_0^x U(y)dy$, and that $\int_0^1 Ku(x)\,u(x)dx = \int_0^1 U(x)^2dx$.
3. Use this identity to show that $0$ is not an eigenvalue of $K$ — i.e., $N(K)=\{0\}$.
4. Show that there is no constant $c>0$ such that $c\|u\|\le \|Ku\|$. Explain why this implies $K^{-1} \not\in \mathcal B(\mathcal H)$. (Hint: consider the sequence $u_n(x) = \sqrt{2} \cos(n\pi x)$.) The point here is that $\lambda=0$ is not an eigenvalue of $K$, but is in the spectrum of $K$.

Assignment 12 - Due Wednesday, December 5, 2018.

• Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and my notes on example problems for distributions.
• Do the following problems.

1. Section 3.4: 2(d) (You may use problem 4 from Assignment 11.)

2. Section 4.1: 4, 7

3. Section 4.2: 1, 3, 4

4. Section 4.3: 3

5. Let $Ku(x)=\int_0^1 k(x,y)u(y)dy$, where $k(x,y)$ is defined by $k(x,y) = \left\{ \begin{array}{cl} y, & 0 \le y \le x\le 1, \\ x, & x \le y \le 1. \end{array} \right.$
1. Show that $0$ is not an eigenvalue of $K$.
2. Show that $Ku(0)=0$ and $(Ku)'(1)=0$.
3. Find the eigenvalues and eigenvectors of $K$. Explain why the (normalized) eigenvectors of $K$ are a complete orthonormal basis for $L^2[0,1]$.

6. 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. Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a self-adjoint Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
3. Use (b) and the spectral theory of compact operators to show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$.

Updated 11/26/2018.