Mock exam

Problem 1¶

For $0\leq r < 1$ we define $P_r(\theta) = \sum_{n = -\infty}^{+\infty}r^{|n|}e^{in\theta}$ for $\theta \in [-\pi,\pi]$.

1. Prove that the series defining $P_r(\theta)$ converges absolutely and uniformly on $[-\pi,\pi]$.
2. Prove that $P_r(\theta)$ is smooth.
3. The Laplacian operator in polar coordinate is given by $\Delta f = \frac{\partial^2 f}{\partial r^2}+\frac{1}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}$, show that $\Delta P_r(\theta) = 0$ on $\mathbb{D}\setminus \{0\}$.
4. Let $f$ be a continuous function on the circle and $\frac{d^k}{d\theta^k}[f]$ be its $k$-th derivative in sense of distribution. Note that this gives a $2\pi$-periodic distribution.
   • Find its Fourier transform.
   • Show that $\frac{d^k}{d\theta^k}[f]*P_r(\theta)$ is harmonic on $\mathbb D$.

Problem 2¶

Let $\Pi(x)$ be the function $\Pi(x) = \begin{cases}1,-\frac{1}{2}<x<\frac{1}{2}\\ 0, else \end{cases}$ on $\mathbb{R}$.

1. Calculate $\mathcal F\Pi, \Pi*\Pi$ and $\mathcal F(\Pi*\Pi)$.
2. Prove $\sum_{n = -\infty}^{+\infty}\frac{1}{(n+\frac{1}{4})^2} = 2\pi^2$. (Hint: Apply the Poisson summation formula to 1).

Problem 3¶

1. Let $\mathcal S(\mathbb{R}^d)$ be the space of Schwartz functions on $\mathbb{R}^d$. Show that for $f\in \mathcal S(\mathbb{R}^d)$, there exists a unique $u\in \mathcal{S}(\mathbb{R}^d)$ such that $\Delta u + u = f$.
2. For real $a>0$, define the operator $(-\Delta)^af(x) = \int_{\mathbb{R}^d} (2\pi |\xi|)^{2a} \mathcal Ff(\xi) e^{2\pi i \langle \xi,x\rangle}d\xi$. Show that $(-\Delta)^af=(-\Delta f)^a$ for an integer $a$.

Errordum.

• In 1 the Δ should be $-\Delta$.
• In 2 the right hand side should be $[-\partial_{11}+\cdots+\partial_{nn}]\cdots[-(\partial_{11}+\cdots+\partial_{nn})] f$, $a$ times.

Problem 4¶

Let $f,g$ be real functions on $\mathbb{Z}/(N)$. The discrete convolution of $f$ and $g$ is given by $f*_Ng(m) = \sum_{k = 0}^{N-1}f(m-k)g(k)$.

1. Prove that $f*_Ng = g*_N f$.
2. Prove the convolution theorem $\widehat{f*_N g} = \hat{f}\hat{g}$, where $\hat{f}$ denotes the discrete Fourier transform on $Z/(N)$.
3. Show that one needs at most $O(N\log N)$ operations to calculate convolution on $Z/(N)$. (Hint: Convolution can be calculated by the inverse Fourier transform of $\hat{f}\hat{g}$, and one has Fast Fourier Transform algorithm to calculate Fourier transform efficiently.)

Errordum.

• The definition of $f*_Ng$ should be replaced to $\frac{1}{N}\sum f(m-k)g(k)$ to make it compatible to the normalization of discrete Fourier transform. Otherwise 2 should be $N \hat{f}\hat{g}$.

Problem 5¶

Let $f(x)$ be a Schwartz function. Calculate the Fourier transform of $f(x)\cos(ax)$ for $a>0$.

Problem 6¶

The ζ function is defined for $s>1$ by $\zeta(s) = \sum_{n =1}^{\infty}\frac{1}{n^s}$. The Γ-function is defined for $s>0$ by $\Gamma(s) = \int_{0}^{\infty}e^{-x}x^{s-1}dx$. The θ-function is given by $\theta(s) = \sum_{n = -\infty}^{+\infty}e^{-\pi n^2s}$.

1. Verify the functional equation $s^{-\frac{1}{2}}\theta(\frac{1}{s}) = \theta(s)$ whenever $s>0$.
2. Prove the identity $\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s) = \frac{1}{2}\int_{0}^{\infty}t^{\frac{s}{2}-1}(\theta(t)-1)dt$ when $s>1$.