# Math 641-600 — Fall 2016

## Assignments

Assignment 1 - Due Tuesday, September 6, 2016.

• Read sections 1.1-1.4
• Do the following problems.
1. Section 1.1: 3(c), 5, 7(a), 8
2. Section 1.2: 1(a,b), 8, 9, 10(a,b)
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 (i) $\min_{u\in U}\|v-u\| = \|v-p\|$ or (ii) $v-p\in U^\perp$, then it satisfies both (i) and (ii). Moreover, if $p$ exists, it is unique.
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 3 above.)
2. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

Assignment 2 - Due Tuesday, September 13, 2015.

• Read the notes on Adjoints & Self-adjoint Operators, and sections 2.1 and 2.2 in Keener.

• Do the following problems.
1. Section 1.3: 2, 3
2. Section 1.4: 4
3. Let $A = \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 1\\ -1 & 1 & 1\end{pmatrix}$. Find the $QR$ decomposition of $A$.
4. Let $\{v_1,\ldots,v_m\}$ be a set of linearly independent vectors in $\mathbb R^n$, with $m < n$, and let $A := [v_1\ \cdots \ v_m]$; that is, $A$ is an $n\times m$ matrix having the $v_j$'s for columns.
1. Use Gram-Schmidt to show that $A=QR$, where $Q$ is an $n\times m$ matrix having columns that are orthonormal and $R$ is an invertible, upper triangular $m\times m$ matrix.
2. 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 $\mathbf x_{min} = R^{-1}Q^\ast \mathbf y$.

5. Let $A$ be an $n\times n$ matrix and let $p_A(\lambda) = \det(A-\lambda I)$. Show that if $A'$ is similar to $A$, then $p_{A'}=p_A$. In addition, show that the trace of $A$, $\text{Tr}(A)=\sum_{j=1}^n a_{jj}$, satisfies $\text{Tr}(A')=\text{Tr}(A)$ and that $\det(A)=\det(A')$.

6. (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: One direction is trivial. To show that $\max_U \sum_{j=1}^k \langle Au_j,u_j \rangle \le \sum_{j=1}^k \lambda_j$ do the following. Put $A$ in diagonal form. Without loss of generality, one may assume that $\lambda_k\ge 0 \ge \lambda_{k+1}$. Choose a matrix $B$ with these properties: for all $x$, $x^TBx\ge 0$, $x^TBx\ge x^TAx$, and $\text{Tr}(B) = \lambda_1+\lambda_2+\cdots +\lambda_k$. Add vectors to the $u_j$'s to get an o.n. basis for $\mathbb R^n$. Use the invariance of the trace proved in the previous problem to complete the proof.)

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. Show that U is diagonalizable. (Hint: follow the proof for the self-adjoint case. The key to this problem is observing that $Ux=\lambda x$ if and only if $U^*x = \bar \lambda x$.)

Assignment 3 - Due Tuesday, September 20, 2016.

• Read sections 2.1, 2.2.1 and 2.2.2 and the notes on Banach Spaces and Hilbert Spaces .
• Do the following problems.
1. Section 2.1: 3, 5

2. 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?

3. 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.

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. 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 $\|f\|_{H^1[0,1]}^2 := \int_0^1\big( |f(x)|^2 + |f'(x)|^2\big)dx$.

Assignment 4 - Due Thursday, September 29, 2016.

Assignment 5 - Due Thursday, October 8, 2016.

Assignment 6 - Due Friday, October 21, 2016.

• Read section 2.2.7 and the notes on Splines and Finite Element Spaces.
• Do the following problems.

1. Section 2.2: 14 (Hint: assume that Fubini's theorem holds.)

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

3. Let $f$ be a piecewise smooth, continuous $2\pi$ periodic function having a piecewise continuous derivative, $f'$. Show that $\int_0^{2\pi} f'(x) e^{-inx}dx = in\int_0^{2\pi} f(x) e^{-inx}dx.$ (Don't forget to deal with the (possible) discontinuities in $f'$ when you integrate by parts.) Use this to show that we may interchange sum and derivative to obtain the Fourier series for $f'$. That is, if $f(x)=\sum_{-\inty}^{\infty} c_n e^{inx}$, then $f'(x) = \frac{d}{dx} \big\{\sum_{-\infty}^{\infty} c_n e^{inx} \big\}=\sum_{-\infty}^{\infty} c_n \frac{d}{dx}e^{inx} = \sum_{-\infty}^{\infty} inc_n e^{inx}$

Use this to show that the sine/cosine form of the result is $f'(x) = \sum_{n=1}^\infty n(a_n\cos(nx) -b_n \sin(nx))$, where the $a_n$'s and $b_n$'s are Fourier coefficients of the Fourier series for $f$.

4. Use the previous problem to show that if $f$ is a piecewise smooth, continuous, $2\pi$-periodic function having a piecewise continuous derivative $f'$, then the Fourier series for $f$ converges uniformly to $f$. (Hint: Note that since $f'\in L^2[0,2\pi]$, the series $\sum_{n=-\infty}^\infty n^2|c_n|^2$ is convergent. Show that $\sum_{n=-\infty}^\infty |c_n|$ is convergent, then apply the Weierstrass M-test to obtain the result.)
5. In class we showed that if $f(x) = \left\{ \begin{array}{cl} 1 & 0 \le x \le \pi, \\ -1 & -\pi \le x< 0 \end{array} \right.$, then the Fourier series for $f$ is given by $f(x)=\sum_{n\ \text{odd}} \frac{4}{n \pi}\sin(nx) = \sum_{k=1}^\infty \frac{4}{(2k-1) \pi}\sin\big((2k-1)x\big)$. Using this series, find the Fourier series for $F(x) =|x|$, $|x|\le \pi$. (You will need to compute just one integral.)

Assignment 7 - Due Friday, October 28, 2016.

Assignment 8 - Due Friday, November 4, 2016.

Assignment 9 - Due Friday, November 11, 2016.

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

1. Section 3.4: 2(b)

2. Let H0 be the inner product space defined problem 3, HW7. This becomes a Hilbert space when we allow $f'$ to be in $L^2[0,1]$.
1. Recall that by problem 5, HW3, if we have $f\in H_0$, then $\|f\|_{C[0,1]} \le \|f\|_{H_0}$. (Assume this holds for $f$ such that $f'$ is in $L^2[0,1]$.) Show that $\Phi_y(f) := f(y)$ is a bounded linear functional on H0.
2. Show that for all $f\in H_0$ there is a function $G_y$ in $H_0$ for which $f(y) = \langle f, G_y\rangle$. ($G_y$ is called a reproducing kernel and H0 is a reproducing kernel Hilbert space. It is also a Green's function for boundary value problem in problem 4, HW7.)

3. Let $L$ be a bounded operator on a Hilbert space $H$. Show that the closure of the the range of $L$ satisfies $\overline{R(L)} = N(L^\ast)^\perp$. (Hint: Follow the proof of the Fredholm alternative, which is just the special case where $R(L)$ is closed.)

4. A sequence {fn} in a Hilbert space H is said to be weakly convergent to f ∈ H if and only if lim n → ∞ < fn,g> = < f,g> for every g∈H. When this happens, we write f = w-lim fn. For example, if {φn} is any orthonormal sequence, then φn converges weakly to 0. You are given that every weakly convergent sequence is a bounded sequence (i.e. there is a constant C such that ||fn|| ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space H. If fn converges weakly to f, then Kfn converges to Kf — that is, lim n → ∞ || Kfn - Kf || = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of {fn}, say {fnk}, such that || Kfnk - Kf || ≥ ε for all k. Use this and the compactness of K to arrive at a contradiction. We remark that the converse is also true. If a bounded linear operator $K$ maps weakly convergent sequences into convergent sequences, then $K$ is compact.

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

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?

Assignment 10 - Due Friday, November 18, 2016.

Assignment 11 - Due Friday, December 2, 2016.

• Read sections 3.6, 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1. and my notes on Examples problems for distributions.
• Do the following problems.

1. Section 4.1: 1(b), 4, 6
2. Section 4.2: 1, 4, 8
3. Section 4.3: 3
4. 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. Let $D$ be an open subset of a Banach space $V$. We say that a map $G:D\to V$ is Lipschitz continuous on $D$ if and only if there is a constant $0<\alpha$ such that $\|G[u]-G[v]\|\le \alpha \|u-v\|$. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
3. Show that $F$ has a fixed point in $B_1$.

5. Let $L$ be in $\mathcal B (\mathcal H)$.
1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
2. Let $|\lambda| \|L\|<1$. Show that $\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^n \|L\|^n}{1 - |\lambda| \|L\|}.$
3. Let $L$ be as in problem 6, HW8. Use the bound on $\|L\|$ in 6(b) of this problem to estimate how many terms of the Neumann expansion it would require to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.2$.

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 11/20/2016.