Math 658-600 Assignments — Fall 2012

Assignment 1 -Due Friday, 9/7/12.

• Read sections 1.3, 2.1-2.3.
• Problems
  1. Let f be in L¹(R) and define the translation operator T_uf(x) = f(x-u). The aim of this problem is to show that lim_{u → 0} ‖f - T_uf‖₁ = 0.
     1. Show that for all f in L¹(R) the translation operator satisfies ‖T_uf‖₁ = ‖f‖₁.
     2. Let g:R → R be continuous and have compact support. Show that for ε > 0 there is a δ > 0 such that
        ‖g - T_ug‖₁ < ε, provided |u| < δ.
     3. Recall that a function in L¹(R) can be approximated arbitrarily closely by a compactly supported, continuous function. Use this to show that
        lim_{u → 0} ‖f - T_uf‖₁ = 0.
     4. Give an example to show that this (the limit above) fails in L^∞(R)

  2. Verify the Fourier series for the functions in entries 1, 5, 6, 13, 18, 19 of Table 1, p. 26-28.
  3. Consider the trig polynomial T(x) = ∑_|n|≤N d_n e^inx.
     1. Show that the Fourier coefficients of T are just the d_n's for |n|≤ N and 0 for |n|> N.
     2. Show that if f has the Fourier series ∑_n c_n e^inx, then
        (2π)^-1 ∫_−π^π f(x)T(x)dx = ∑_|n|≤N c_nd_n.
        
     3. Show that (2π)^-1 ∫_−π^π |T(x)|²dx = ∑_|n|≤N |d_n |², so Bessel's inequality becomes an equation for trig polynomials.
     4. Show that if f has the Fourier series ∑_n c_n e^inx, then
        (2π)^-1 ∫_−π^π|f(x) − T(x)|²dx = (2π)^-1 ∫_−π^π|f(x)|²dx − ∑_|n|≤N |c_n|² + ∑_|n|≤N |d_n − c_n|²

Assignment 2 — Due Monday, 9/17/2012.

• Read sections 2.4-2.6, 7.1-2.
• Problems
  1. Section 2.2: 7
  2. Section 2.3: 5
  3. Section 2.6: 1
  4. Let p=1 or ∞. Suppose that f ∈ L^p and g ∈ L¹ are both $2\pi$ periodic functions. Show that the convolution
     f∗g(θ) := (2π)^-1 ∫₀^2π f(θ − φ)g(φ)dφ
     is in L^p and that ‖f∗g‖_p ≤ (2π)^-1‖f‖_p ‖g‖₁.
     Bonus: Show that the inequality above holds for all 1 ≤ p ≤ ∞. (You will need to use both the Fubini-Tonelli Theorem and Hölder's inequality.)

  5. The Cesàro means of the partial sums s_n(θ) of a Fourier series a function f are defined to be
     σ_n(θ) = (n+1)⁻¹∑_0≤k≤n s_k(θ)
     1. Show that σ_n(θ) = ∑_{|k|≤ n}(1 − |k|∕(n+1)) c_ke^ikθ.
     2. The Fejér kernel F_n(u) is defined to be the Cesàro mean of the Dirichlet kernels D_k(u). Show that
        F_n(u) = [2π(n+1) sin²(½u)]^{− 1} sin²(½(n+1)u).
        
     3. Show that σ_n(θ) is also given by the convolution
        σ_n(θ) = 2πF_n ∗f(θ).
        
     4. Show that if f is a 2π periodic continuous function, then σ_n converges uniformly to f.

Assignment 3 — Due Monday, 9/24/2012.

• Read sections 7.3, 7.5, 7.6
• Problems
  1. Section 7.1: 2, 3(a,b) (just do f∗f in (a)), 5, 7
  2. Section 7.2: 4, 7, 12(a), 13(b), 14 (Note: Problem 14 has a misprint in it. The Hermite functions are defined on the bottom of p. 186 and their properties are discussed on p. 187.)
  3. Suppose that f ∈ C¹(R), and that both f and f′ are bounded on R, and that g ∈ L¹(R). Show that f∗g ∈ C¹(R) and that (f∗g)′ = f′∗g.

Assignment 4 — Due Wednesday, 10/3/2012.

• Read my Notes on the Discrete Fourier Transform.
• Problems
  1. Section 7.1: 8
  2. Section 7.2: 13(a)
  3. Section 7.3: 6, 8, 9, 10 (There is a misprint in 10. f′ + cf = 0 should be f′ + cxf=0.)
  4. Section 7.5: 2
  5. Suppose that φ: Rⁿ → C and that φ ∈ L¹(Rⁿ) ∩ C₀(Rⁿ). Furthermore, suppose that $\hat \varphi(\xi) > 0$. Prove the following result: If $\{x_j\}_{j=1}^N$ is a finite set of distinct points in Rⁿ, then the matrix $A$ with entries $A_{j,k} = \varphi(x_j - x_k)$ is Hermitian and positive semi-definite. (Note: $\hat \varphi$ is in L¹(Rⁿ). Prove it for a 5 pt. bonus.)

  6. Let f(t) be band-limited, with its Fourier transform f^(ω) (f "hat") having support in [−2π, 2π]. The Nyquist rate for f is 2π/π = 2, and the Nyquist frequency is ½·2 = 1.
     1. Show that if f is sampled at the Nyquist frequency, instead of the Nyquist rate, then the sampling theorem doesn't recover f. Instead, it gives another band-limited function, f_alias. Show that on [−π, π],
        f^_alias(ω) = f^(ω − 2π) + f^(ω) + f^(ω + 2π),
        and that f^_alias = 0 otherwise.
     2. Strictly speaking, the function e^iω₀t doesn't have a Fourier transform in the usual sense. However, if we allow tempered distributions, then it does have one,2π δ(ω − ω₀). Another example of a function with a distributional Fourier transform is cos(ω₀t). Its Fourier transform is 2π(δ(ω − ω₀) + δ(ω + ω₀)/2. Keeping this in mind and referring to the previous part, consider the band-limited function
        f(t) = sin(5πt/3+π/6),
        whose angular frequencies ±5π/3 are in the band [−2π, 2π]. What is f_alias(t) in this case? Use your favorite software to make a plot showing f(t), f_alias(t) on the interval − 6 ≤ t ≤ 6, and also use circles or x's to indicate samples of these two functions at integer values of t.

Assignment 5 — Due Friday, 10/12/2012.

• Read the article Radon Inversion — Variations on a Theme, by Robert S. Strichartz, Amer. Math. Monthly, 89 (1982), 377-384.
• Problems
  1. Section 7.5: 3, 6(b)
  2. Prove the three formulas listed at the end of section 2 on p. 3 of the Notes on the Discrete Fourier Transform.
  3. Prove: Let L be a linear filter with impulse response h, where h is in L¹(R) and is continuous. Then, if L is causal, h(t) = 0 for all t < 0. (This completes the proof of Proposition 3.1 in my Notes on Filters.)
  4. Let {x₁, x₂, ..., x_N} be a set of distinct points in R. Show that the set of functions
     {exp(ix₁t), exp(ix₂t), ..., exp(ix_Nt)}
     is linearly independent. (Hint: Consider a Fourier transform pair $f, \hat f$ such that both functions are in L¹, and hence also in C₀(R). Then,
     c₁exp(ix₁ξ) + c₂exp(ix₂ξ) + ... + c_N exp(ix_Nξ) = 0, for all ξ,
     is equivalent to
     c₁f(x₁) + c₂f(x₂) + ... + c_Nf(x_N) = 0 for all f in L¹.
     Choose the pair $f$ and $\hat f$ wisely.)
  5. (This problem is a follow-up to problem 5, HW 4.) Suppose that φ: R → C and that φ ∈ L¹(R) ∩ C₀(R). Furthermore, suppose that $\hat \varphi(\xi) > 0$ on a set of positive measure. Use the previous problem to prove the following result: If $\{x_j\}_{j=1}^N$ is a finite set of distinct points in R, then the matrix $A$ with entries $A_{j,k} = \varphi(x_j - x_k)$ is strictly positive definite. You may use the fact that $\hat \varphi$ is in L¹(R).
  6. Let f: R → C be continuous and have support in the interval [0, A], and let n be a positive integer. If y_j = f(jΔt), where Δt = A/n, then show that \[ \hat f(kΔω) \approx \hat y_k \Delta t, \quad \Delta \omega = \frac{2\pi}{A}. \]
  7. Let n be a positive integer. Consider the space of continuous piecewise functions on [0,1] with corners at j/n, j = 0, $\ldots$, n. These are linear splines and have a basis {T(x − j/n}, where T is the tent function, \[ T(x):= \left\{ \begin{array}{cc} 1 - |x| & |x|\le 1 \cr 0 & |x| > 1 \end{array} \right.. \]
     1. Let Δx = 1/n. Recall that if f is in C[0,1], then \[ s_f(x) = \sum_{j=0}^n f(j\Delta x)T\big(n(x - j\Delta x)\big) \] interpolates f on the interval [0, 1] at the points jΔx. Use this to show that if f ∈ C⁽²⁾[0,1], then \[ \|f-s_f\|_{C[0,1]} \le \|f''\|_{C[0,1]} n^{-2}. \]
     2. Show that the trapezoidal rule for $\int_0^1 f(x)dx$ is just $\int_0^1 s_f(x)dx$, and then use the previous part to show that, for f ∈ C⁽²⁾[0,1], the error for the trapezoidal rule is $\|f''\|_{C[0,1]} n^{-2}$.
     3. Let $\{\hat y_k\}$ be the DFT for a 2π periodic function f that is twice continuously differentiable on R, and let $\{c_k\}$ be the Fourier coefficients for f. Show that the error $|c_k - \hat y_k/n|$ is at worst of order $(k/n)^2$. (We can do better!)

Assignment 6 — Due Friday, 11/2/2012.

• Read sections 9.1-9.4 in the text.
• Problems
  1. Find the Radon transform of the function of the characteristic function χ_D for the unit disk, D.
  2. Let $f\in \mathscr S$ (Schwartz space) and recall that we defined $ \|f\|_k:=\max_{\,|\,\alpha|\le k,\,|\beta|\le k}\|f\|_{\alpha,\beta}. $
     1. Show that $\|f\|_k$ is a seminorm for $\mathscr S$.
     2. The following identity will be needed in the next part. Show that if $x,y,z$ are positive and if $x \le y + z $, then \[ \frac{x}{1+x}\le \frac{y}{1+y} + \frac{z}{1+z}. \] Hint: $f(t) = \frac{t}{1+t}$ is increasing, so $f(x)\le f(y+z)$.
     3. Let $\rho(f) := \sum_{k=0}^\infty \frac{\|f\|_k }{1+\|f\|_k}2^{-1-k}$ and let $d(f,g):=\rho(f-g)$. Show that $d(f,g)$ is a metric for $\mathscr S$. You will need to use the inequality in (b).
  3. Let $f_j$ be a sequence of functions in $\mathscr S$ and let $\rho$ be as in problem 2. Show that for each fixed $k$ we have $\lim_{j\to \infty}\|f_j\|_k=0$ (not necessarily uniformly in $k$) if and only if $\lim_{j\to \infty} \rho(f_j)=0$.
  4. Use induction on the number of variables to prove the multivariate product rule below. (Multi-index notation is used throughout. See the Wikipedia entry for Multi-index notation.) \[ \partial^\alpha (fg) = \sum_{0\le \beta \le \alpha}\binom{\alpha}{\beta}\partial^\beta f \partial^{\alpha -\beta}g. \]
  5. Section 9.4: 3(a) (Use the previous problem.)

Assignment 7 — Due Monday, 11/12/2012.

• Read the Notes on Scattered-Data Radial Function Interpolation.
• Problems
  1. Section 9.4: 4, 6, 14(b), 15
  2. The setting for this problem is $\mathbb{R}^2$. Consider the $f(r) = 1/r$.
     1. Show that $F(\phi):=\langle f,\phi \rangle$ defines a tempered distribution - i.e., it is in $\mathscr{S}'$.
     2. Consider the sequence of functions $f_k(r) = \frac{e^{-r/k}}{r}$, all of which are in $L^1(\mathbb R^2)$. Let $F_k(\phi) = \langle f_k,\phi \rangle$ be the tempered distributions corresponding to $f_k$. Show that $F_k \to F$ in $\mathscr{S}'$.
     3. Show that \[ \hat{f}_k(\xi) = \frac{2\pi}{\sqrt{k^{-2}+|\xi|^2}}. \] Hint: First show that $\hat{f}_k(\xi) = \int_0^{2\pi}(k^{-1}+i|\xi| \cos(\theta))^{-1}d\theta$. Then evaluate the integral using contour integration techniques. (If you haven't had contour integration, then see me.)
     4. Show that $\hat{f}(\xi)=\displaystyle{\frac{2\pi}{|\xi|}}$.
     5. Let $\phi$ be in $\mathscr S$ and recall that, in the discussion of the Radon transform, we used the Riesz transform $I_1[\phi](x)= (\phi*f)(x)/(2\pi)$, $x \in \mathbb{R}^2$. Find the Fourier transform of $I_1[\phi]$.

Assignment 8 — Due Wednesday, 11/28/2012.

• Read the Notes on Spherical Harmonics.
• Problems
  1. Let $\{c_n\}_{n\in \mathbb Z}$ satisfy $|c_n|\le (1+|n|)^m$ for $n\not=0$ and $m$ some fixed positve integer. Show that the Fourier series $\sum_{n\in \mathbb Z}c_ne^{inx}$ defines a $2\pi$-periodic tempered distribution.
  2. Consider the function $\Phi(r) := \max(1-r,0)$. On $\mathbb R$, this is just the linear "tent" function and is positive definite. By finding its Fourier transform in $\mathbb R^3$, show that $\Phi(r)$ is not positive definite in $\mathbb R^3$.
  3. Suppose that $\Psi( r )$ is a positive definite radial function on $\mathbb R^n$. Show that it is positive definite on $\mathbb R^d$ for all $1\le d\le n$.
  4. Let $\kappa(x,y)$ be a strictly positive definite kernel on ${\mathbb R}^n$, say. Suppose that $\mathscr F^0$ is the set of all $f=\sum_{\xi \in X}a_\xi \kappa(\cdot,\xi)$, where $X$ is a pairwise distinct subset of ${\mathbb R}^n$. Show that for $f,g\in \mathscr F^0$ the form below is bilinear in $f$ and $g$. \[ \langle f,g\rangle = \sum_{\xi\in X,\,\eta\in Y} a_\xi b_\eta \kappa(\xi,\eta). \]
  5. Let $\mathscr F$ be a reproducing kernel Hilbert space (RKHS), with kernel $\kappa(\cdot,\cdot)$. Assume the functions in $\mathscr F$ are defined on $\mathbb R^n$. Let $X$ be a finite set of distinct points in $\mathbb R^n$ and define $V_X = {\rm span}\{\kappa(\cdot, \xi); \xi\in X\}$.
     1. Let $f$ be in $\mathscr F$. Recall that the interplant $I_X f$ is the orthogonal projection of $f$ onto $V_X$. Use this and the reproducing property of $\kappa$ to show that \[ |f(x)-I_Xf(x)|\le \|f-I_Xf\|_{\mathscr F} \|\kappa(\cdot,x)-\sum_{\xi\in X}c_\xi \kappa(\cdot, \xi)\|_{\mathscr F}, \] where the coefficients $\{c_\xi\}_{\xi\in X}$ are arbitrary.
     2. Show that $\|f-I_Xf\|_{\mathscr F} \le \|f\|_{\mathscr F}$.
     3. Define the power function by $\mathcal P_X(x) : = \min_{\,c_\xi,\, \xi\in X}\|\kappa(\cdot,x)-\sum_{\xi\in X}c_\xi \kappa(\cdot, \xi)\|_{\mathscr F}$. Use the previous parts to show that for all $x\in \mathbb R^n$ we have that \[ |f(x)-I_Xf(x)|\le \|f\|_{\mathscr F}{\mathcal P}_X(x). \]
  6. Let $A$ be an $n\times n$ self-adjoint, real matrix that is positive definite. Define the condition number of $A$ to be $c(A):=\lambda_{max}/\lambda_{min}$ — i.e., the ratio of maximum and minimum eigenvalues.
     1. Consider the equation $Ax = b$. Suppose that there is an error $\Delta b$ in $b$, resulting in an error $\Delta x$ in $x$. Show that \[ c(A)^{-1}\frac{\| \Delta b \|_2}{ \|b\|_2} \le \frac{\|\Delta x\|_2}{\| x \|_2} \le c(A)\frac{\| \Delta b \|_2}{ \|b\|_2}. \]
     2. Let $\varepsilon >0$ and consider the matrix $A = \begin{pmatrix} 1 & 1\\ 1 & 1+\varepsilon \end{pmatrix}$. Show that $c(A) \approx 4\varepsilon^{-1}$.
     3. Let $b=(1\ 1)^T$ and note that $x=A^{-1}b=(1 \ 0)^T$. Suppose that we know that the error in $b$ is roughly $\Delta b= 10^{-4}(1 \,\ -\!1)^T$. (This means that the entries in $b$ are accurate to about $10^{-4}$.) If $\varepsilon = 4\times 10^{-6}$, show that $\|\Delta x\|_2\approx 10^{2}$. With this in mind, how reliable are the entries in $x$?
  7. Suppose that $\Phi$ is a strictly positive definite radial function in $\mathbb R^n$ for all $n$ — i.e., $\Phi( r )$ is on order 0 RBF.
     1. Show that $\Phi( r )>0$ for $0 < r $.
     2. Show that $\Phi$ is in $C^\infty((0,\infty))$. (Actually, it is analytic.)

Assignment 9 — Not to be handed in.

• Problems
  1. Let $L_\pm$ be the raising and lowering operators in the Notes on Spherical Harmonics. Use them to show that, for $m\ge 0$, if $Y_{\ell,m}(\theta,\varphi) = Q_{\ell,\,m}(\cos(\theta))\sin^m(\theta)e^{im\varphi}$, where $Q_{\ell,\,m}(t)$ is a polynomial in $t$, then $Q_{\ell,\,m-1} = C((1-t^2)Q'_{\ell,\, m}-2m t Q_{\ell,\,m})$ and, in addition, that $ Q_{\ell,\,m+1} = C Q_{\ell,\,m}'(t)$. Use these formulas along with $Y_{\ell,\,\ell}(\theta,\varphi)=C\sin^\ell(\theta)e^{i\ell \varphi}$ to show that the $Y_{\ell,\, m}$ have the form in Theorem 3.1, up to normalization constants. ("C" is a generic constant.)

  2. Use Theorem 3.1 in the Notes on Spherical Harmonics to expand the function $f(\theta,\varphi) = 2 - \cos^2(\theta)+ \cos(\theta)\sin(\theta)^2\sin^2(\varphi)$ in spherical harmonics.

  3. Recall that a function $f$ in $\mathbb R^3$ is said to be harmonic if and only if $\Delta_{\mathbb R^3}f = 0$, and also that a harmonic polynomial homogeneous of degree $\ell$ is related to a spherical harmonic $Y_\ell$ via $p(\mathbf x)=r^\ell Y_\ell(\mathbf x/r)$. (Here $\Delta_{\mathbb S^2}Y_\ell + \ell(\ell+1)Y_\ell =0$.) Find a basis for harmonic polynomials homogeneous of degree $\ell=3$ using Theorem 3.1 in the Notes on Spherical Harmonics.

  4. Fix $m\ge 1$. Show that the polynomials $\{P^{(m)}_\ell : \ell\ge m \}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(t)\overline{g(t)}(1-t^2)^m dt$. (Hint: use Theorem 3.1.)

  5. For the rotation matrix $R$ below, find these: the Euler angles, the angle of rotation, and the axis of rotation. \[ R =\begin{pmatrix} \frac{2}{3} & \frac{2}{3} &-\frac{1}{3}\\ -\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix}. \]
  6. Let $A$ be an n×n matrix such that $A^3=-A$. Show that $e^{\theta A} = I + \sin(\theta) A + (1-\cos(\theta))A^2$. Using the axis of rotation $\mathbf \omega$ and angle of rotation $\theta$ for $R$ found in the previous problem, find $A$ such that $R=e^{\theta A}$. (Hint: recall that $A\mathbf x = \mathbf \omega \times \mathbf x$ and that $R\mathbf x = \mathbf x + \sin(\theta)\mathbf \omega \times \mathbf x+ (1-\cos(\theta))\mathbf \omega\times (\mathbf \omega \times \mathbf x)$.)

  7. Let $\mathbf 1 = \begin{pmatrix} 1&0\\0&1\end{pmatrix}$, $\mathbf i = \begin{pmatrix} i&0\\0&-i\end{pmatrix},\ \mathbf j = \begin{pmatrix} 0&1\\ -1&0\end{pmatrix},\ \mathbf k = \begin{pmatrix} 0&i\\i&0\end{pmatrix}.\ $ For $q_0\in \mathbb R$ and $\mathbf q\in \mathbb R^3$, a quaternion $Q = q_0\mathbf 1 + \mathbf q$, where $\mathbf q = q_1\mathbf i+q_2\mathbf j + q_3\mathbf k$ is the "pure" part of $Q$. The set of quaternions is denoted by $\mathbb H$. Also, $SU(2)$ comprises all $U\in \mathbb H$ such that $u_0^2 + |\mathbf u|^2 = 1$.
     1. Show that $PQ = (p_0q_0 - \mathbf p\cdot \mathbf q)\mathbf 1 + q_0\mathbf p + p_0\mathbf q+\mathbf p \times \mathbf q$.

     2. Let $U=u_0\mathbf 1 + \mathbf u$ be in $SU(2)$ and let $X=\mathbf x$ be a pure quaternion representing a point in $\mathbb R^3$. In addition, let $\overline{U} = u_0\mathbf 1 - \mathbf u$. Show that \[ UX\overline{U} = \mathbf x + 2u_0\mathbf u \times \mathbf x+ 2\mathbf u \times (\mathbf u\times x). \]
     3. If $u_0=\cos(\psi)$ and $\mathbf u = \sin(\psi)\mathbf \omega$, where $\mathbf \omega \in S^2$, then show that \[ UX\overline{U} = \mathbf x + \sin(2\psi)\mathbf \omega \times \mathbf x+ (1-\cos(2\psi))\mathbf \omega \times (\mathbf \omega\times \mathbf x). \]
     4. Find all quaternions $U$ that correspond to the rotation matrix $R$ in problem 5. For each of them, what is $\psi$?