Math 641-600 — Fall 2023

• The problems numbers refer to sections in the text. Inner Products. Read sections 1.1-1.3, 2.1 and the notes on and my notes on coordinates and bases, on inner products and on Banach spaces and Hilbert Spaces

• Do the following problems.
  1. Section 1.1: 4, 5, 8 (Only do the first three, and do them by hand.)
  2. Section 1.2: 1(a), 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 is a vector $p \in U$ that satisfies either $\min_{u\in U}\|v-u\| = \|v-p\|$ or $v-p\in U^\perp$, then it satisifes both. In addition, show that if such a vector exists, then 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 $P$ has the following properties:
        1. $P$ maps $V$ onto $U$.
        2. $P$ is a linear map.
        3. $P$ satisfies $P^2 = P$ and $P^\ast=P$.  ($P$ is called an orthogonal projection. The vector $p$ is the orthogonal projection of $v$ onto $U$.)
        4. $U^\perp= \{w\in V\colon Pw=0\}$ and $V=U\oplus U^\perp$, where $\oplus$ indicates the direct sum of the two spaces. (This and the next exercise are easy, but important.)
        5. $I -P$ is the projection of $V$ onto $U^\perp$, and that the Pythagorean theorem, $\|v\|^2=\|Pv\|^2+\|(I-P)v\|^2$, holds.

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

• Do the following problems.

  1. Section 1.3: 2, 5 (Hint: First show this is true for $n=2$. Then use Lagrange multipliers, with constraints $\|\mathbf u\|=1$, $\|\mathbf v\|=1$ and $\langle \mathbf u,\mathbf v\rangle =0$, to find an appropriate 2 dimensional invariant subspace where $\langle Au,u\rangle +\langle Av,v\rangle$ is maximized.)

  2. Let $\mathcal P_3$ be the polynomials of degree $3$ or less. If $L(u) = 2xu'-u''$, show that $L:\mathcal P_3 \to \mathcal P_3$. Find $L^*$ relative to the inner product $\langle p,q\rangle =\int_0^1 p(x)q(x)$. Also, find the eigenvalues of $L$.

  3. 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 $b=\begin{pmatrix} 1\\ 3\\ 7 \end{pmatrix}$. (See the the sections on minimization problems QR factorization in my notes on Inner products and Norms.)

  4. 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.
     4. Show that from the eigenvectors of $U$ one may select an orthonormal basis for $V$ – i.e., $U$ is diagonalizable. (Hint: follow the proof for self-adjoint case.)

• Read Keener's section 2.1 and the notes on Lebesgue integration.
• Do the following problems.
  1. Section 2.1: 1, also show that $\ell^2$ is a Hilbert space.

  2. 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.

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

• Read the notes on Orthonormal sets and expansions and on Approximation of continuous functions.
• Do the following problems.
  1. Section 2.2: 8(a,b,c,d) (In (d) use $T_n(x)=\cos(n\cos^{-1}(x))$. FYI: the formula for $T_n(x)$ has an $n!$ missing in the numerator.), 9

  2. Show that, in terms of the partial sums in eqn. 2 of my notes, the integral of a simple function ends up being the one in eqn. 3.

  3. This problem is aimed at showing that the Chebyshev polynomials form a complete set in $L^2_w$, which has the weighted inner product \[ \langle f,g\rangle_w := \int_{-1}^1 \frac{f(x)\overline{g(x)}dx}{\sqrt{1 - x^2}}. \]
     1. Show that the continuous functions are dense in $L^2_w$. Hint: if $f\in L^2_w$, then $ \frac{f(x)}{(1 - x^2)^{1/4}}$ is in $L^2[-1,1]$.
     2. Show that if $f\in L^\infty[-1,1]$, then $\|f\|_w \le \sqrt{\pi}\|f\|_\infty$.
     3. Follow the proof given in the notes on Orthonormal Sets and Expansions showing that the Legendre polynomials form a complete set in $L^2[-1,1]$ to show that the Chebyshev polynomials form a complete orthogonal set in $L^2_w$.

  4. Let F(s) = ∫ ₀^∞ e ^{− s t} f(t)dt be the Laplace transform of f ∈ L¹([0,∞)). Use the Lebesgue dominated convergence theorem to show that F is continuous from the right at s = 0. That is, show that
     lim _s↓0 F(s) = F(0) = ∫ ₀^∞f(t)dt

  5. 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.

  6. 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 the notes on Fourier series, and the notes on the discrete Fourier transform, 2.2.7 and the notes on Splines and Finite Element Spaces.
• Do the following problems.
  1. Section 2.2: 14

  2. Compute the Fourier series for the following $2\pi$ functions. In each case sketch three periods of the function to which the series converges.
     1. $f(x) = |x|$, $-\pi \le x \le \pi$. (sine/cosine form)
     2. $f(x) = e^{2x}$, $-\pi \le x \le \pi$. (complex form)
     3. $f(x) = e^{2x}$, $0 \le x \le 2\pi$. (complex form) Compare this series with the one above. Does the FS you get contradict the result in the Lemma from the Notes on Fourier Series?

  3. Use the FS from 6(d) above and Parseval's theorem to sum the series $\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.

  4. The following problem is aimed at showing that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.
     1. Show that the FS for a linear spline $s(x)$ that satisfies $s(-\pi)=s(\pi)$ is uniformly convergent to $s(x)$ on $[-\pi,\pi]$.
     2. Show that such splines are dense in $L^2[-\pi,\pi]$.
     3. Show that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.

  5. 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 $.

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

  6. Let $\mathbf a\in \mathcal S_n$ and consider an n×n matrix $\mathbf A$ whose first column is $\mathbf a=[a_0 \ a_1 \ \cdots \ a_{n-1}]^T$, and whose remaining columns are entries of $\mathbf a$ cyclically permuted. For example, the second colunm is $[a_{n-1} \ a_0 \ \cdots \ a_{n-2}]^T$, third, $[a_{n-2} \ a_{n-1} \ a_0 \cdots \ a_{n-3}]^T$ and so on. $\mathbf A$ is said to be a cyclic or circulant matrix.
     1. Consider the the matrix equation $\mathbf A \mathbf x= \mathbf y$, where $\mathbf x$ and $\mathbf y$ are columns vectors. These vectors may be extended to vectors in $\mathcal S_n$ by repeating the entries $0,\ldots,n-1$ to form a sequence. For $\mathbf x$ this is given by \[ \mathbf x = \{\cdots x_{0} \ x_{1} \ \cdots \ x_{n-1} \ x_0 \ x_1 \ \cdots \ x_{n-1} \ x_{0} \ x_{1} \ \cdots \ x_{n-1}\ \cdots\} \] Show that the equation $\mathbf y = \mathbf A \mathbf x$ is equivalent to the convolution equation $\mathbf y = \mathbf a\star \mathbf x$.

     2. Use the DFT and the convolution theorem to show that the eigenvalues of a cyclic matrix $\bf A$ are the entries in $\widehat {\mathbf a}$, the DFT of $\mathbf a$.

     3. Use your favorite software to find $\widehat {\mathbf a}$, and hence, the eigenvalues of the matrix below. Also, find the corresponding eigenvectors. \[ \begin{pmatrix} 3 &5 &4 &1 \\ 1 &3 &5 &4 \\ 4 &1 &3 &5\\ 5 &4 &1 &3 \end{pmatrix} \]

  7. Consider the space of cubic Hermite splines $S_0^{1/n}(3,1)\subset S^{1/n}(3,1)$ that satisfy $s(0)=s(1)=0$. Show that $\langle u,v\rangle = \int_0^1 u''v''dx$ defines an inner product on $S_0^{1/n}(3,1)$.

Assignment 7 - Due Friday, November 4, 2022.

• Read sections 3.1-3.3, the notes on Bounded Operators & Closed Subspaces and on the projection theorem, the Riesz representation theorem, etc .

• Do the following problems.
  1. Section 2.2: 25, 27(a)

  2. Let $f$ be $2\pi$ periodic, $C^{(m-1)}$ function hving $f^{(m)}$ in $L^2[0,2\pi]$. Show that the FS coefficients for $f$ satisfy $c_k=c^{(m)}_k/(ik)^m, k\ne 0$. In addition, let $S_N$ be the $N^{th}$ partial sum of the Fourier series for $f$. For $m\ge 2$ and $N=n-1$, show that the error in the trapezoidal rule applied to $f$ satisfies $\big|Q_n(f)-\frac{1}{2\pi}\int_0^{2\pi}f(x)dx\big|\le Cn^{-(m-1)}\|f^{(m)}\|_{L^1[0,2\pi]}$, where $C=C(m)$.

Assignment 8 - Due Wednesday, November 16, 2022.

• 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. Let $k(x,y)$ be 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. Section 3.3: 1 (Assume the appropriate operators are closed and that λ is real.)

     2. Section 3.4: 2(b)

     3. Let $k(x,y)$ be 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. 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 $\|L\|_{C[0,1]\to C[0,1]}\le 1$. Bonus (5 pts.): Show that $\|L\|_{C[0,1]\to C[0,1]}=1/2$.

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

     4. 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 $\ell^2$.

     5. 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?

     6. 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}.
        1. Verify the identity < L(u+αv), u+αv> − < L(u-αv), u-αv> = 2α< Lu,v>+2α< Lv,u>, where |α| = 1.
        2. Show that N ≤ ||L||.
        3. Let L be a self-adjoint operator on H, which may be real or complex. Use (a) and (b) to show that N= ||L||. (Hint: In the complex case, choose α so that α< Lu,v> = |< Lu,v>|. For the real case, use $\alpha=\pm 1$, as required.)
        4. Suppose that H is a complex Hilbert space. If L ∈ B(H), then use (a) and (b) to show that
           N ≤ ||L|| ≤ 2N.
           
        5. For the real Hilbert space, H = R², let $L = \begin{pmatrix} 0& 1\\ -1 & 0 \end{pmatrix}. $ Show that $\|L\| = 1$, but $N=0$.

• Read sections 4.1, 4.2, my notes on and my notes on example problems for distributions.

• Do the following problems.
  1. Section 3.4: 6 (The condition in 6 should be $\lambda \mu_i\ne 1$.)

  2. Section 3.5: 2(b)

  3. 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.

  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. 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. Show that the fixed point in the contaction mapping theorem is unique.

  6. (This is a variant of problem 3.4.3 in Keener.) Consider the operator $Ku(x) = \int_{-1}^1 (1-|x-y|)u(y)dy$ and the eigenvalue problem $\lambda u = Ku$.
     1. Show that $K$ is a self-adjoint, Hilbert-Schmidt operator.
     2. Let $f\in C[-1,1]$. If $v= Kf$, show that $-v''=2f$, $v(1)+v(-1)=0$, and $v'(1)+v'(-1)$.
     3. Use the previous part to convert the eigenvalue problem $\lambda u = Ku$ into this eigenvalue problem: \[ \left\{ \begin{align} u''+&\frac{2}{\lambda} u =0,\\ u(1)+&u(-1) =0 \\ u'(1)+ &u'(-1)=0. \end{align} \right. \]
     4. Solve the eigenvalue above to get the eigenvalues and eigenvectors of $\lambda u = Ku$. Show that the eigenvectors form a complete set for $L^2[-1,1]$.
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Updated 9/25/2023.