# Math 641-600 — Fall 2014

## Assignments

Assignment 1 - Due Wednesday, September 10.

• 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 pA(λ) := det(A − λ I) is independent of the choice of E, and so pL(λ) := pA(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
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.
4. Let f(x) : x2, -1 ≤ x ≤ 2. Find f −1(Ej) for Ej = [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.

Assignment 5 - Due Wednesday, October 8.

Assignment 6 - Due Friday, October 31.

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 ∈S1/n(1,0) : s(0)=s(1)=0}. Show that S0 is spanned by φj(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 αjφj(x), where the αj's satisfy Gα = β, with
βj = ⟨ y,φjH0 = ∫01 f(x) φj(x) dx, j=1 ... n-1 and Gkj = ⟨ φj, φkH0.

4. Show that Gkj = ⟨ φj, φkH0 is given by
Gj,j = 2n, j = 1 ... n-1
Gj,j-1 = - n, j = 2 ... n-1
Gj,j+1 = - n, j = 1 ... n-2
Gj,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, ..., 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 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 {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. 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 strongly 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.

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 {φj} corresponding to all eigenvalues λj ≠ 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.