Problem 1 (qiuzhen-qualify 2025-spring)
Section: Analysis

(10 pts) Compute the Fourier transform $\hat u(\xi)$ of $u(x) = \frac{x}{1+x^2}$. (Here we use the following definition for Fourier transform of $L^1$ functions
$$\hat u(\xi) = \int_{\mathbb R} e^{-ix\xi} \, u(x)\, dx$$
and extend to tempered distributions.)

Problem 2 (qiuzhen-qualify 2025-spring)
Section: Analysis

(10 pts) Compute the Fourier series
$$\sum_{n\in\mathbb Z} a_n e^{inx}$$
of the characteristic function $f$ of $[-1,1]$, that is,
$$f(x)=\begin{cases}1 & \text{if } |x|\le 1,\\ 0 & \text{if } 1<|x|\le \pi,\end{cases}$$
and find all points $x\in[-\pi,\pi]$ for which this series converges absolutely.

Problem 3 (qiuzhen-qualify 2025-spring)
Section: Analysis

(10 pts) For all $u_0\in C_c^\infty(\mathbb R)$, we define $u(t,x)\in C^\infty(\mathbb R^2)$ as follows
$$u(t,x)=\frac{1}{\sqrt{2\pi}} \int_{\mathbb R} e^{i(t\xi^2 + x\xi)} \hat u_0(\xi)\, d\xi.$$
Show that for all $x_0\in\mathbb R$, the function $t\mapsto \partial_x u(t,x_0)$ belongs to $L^2(\mathbb R)$ and there exists a constant $c_0>0$ independent of $x_0$ and $u$, such that
$$\int_{\mathbb R} |\partial_x u(t,x_0)|^2\, dt = c_0 \int_{\mathbb R} |u_0(x)|^2\, dx.$$

Problem 4 (qiuzhen-qualify 2025-spring)
Section: Analysis

(10 pts) Let $f:\Omega\to\mathbb C$ be non-constant and holomorphic, where $\Omega\subset\mathbb C$ is an open set containing the closed unit disk $|z|\le 1$. Assume that $|f(w)|=1$ whenever $|w|=1$, show that $f(\Omega)$ contains the open unit disk. (Hint: reduce to the statement that $f(z)=0$ has at least one root in the disk.)

Problem 5 (qiuzhen-qualify 2025-spring)
Section: Analysis

(15 pts) Consider the following second order linear equation for $u=u(x)$:
$$x^2\frac{d^2u}{dx^2} + 2x\frac{du}{dx} + u = 0.$$
(i) Prove that all nontrivial real-valued solutions have infinite number of zeroes on $(1,\infty)$.
(ii) Is it true that all nontrivial real-valued solutions must have finite number of zeroes on $(0,1)$? Prove your answer.

Problem 6 (qiuzhen-qualify 2025-spring)
Section: Analysis

(15 pts) Let $p\in [1,\infty)$ and $\{f_n\}_{n=1}^{\infty}$ a sequence of functions in $L^p(\mathbb R)$ such that $f_n \to f$ a.e. and $f\in L^p(\mathbb R)$.
(i) If $p\in (1,\infty)$, prove that if $\sup_n \|f_n\|_{L^p} < \infty$, then $f_n$ converges to $f$ weakly, i.e., for any $g\in L^q(\mathbb R)$ with $q=\frac{p}{p-1}$,
$$\lim_{n\to\infty} \int_{\mathbb R} f_n g\, dx = \int_{\mathbb R} f g\, dx.$$
(ii) Show that the result in (1) does not hold when $p=1$, $q=\infty$.

Problem 7 (qiuzhen-qualify 2025-spring)
Section: Analysis

(15 pts) Assume that $n\ge 2$, $p\in (1,2)$, and $q\in [p,+\infty]$.
(i) Show that there exists a constant $C$ (may depend on $p,q,n$) such that for all radial function $f\in C_c^\infty(\mathbb R^n\setminus \overline{B})$, there holds:
$$\|f\|_{L^q(\mathbb R^n)} \le C\, \|f\|_{W^{1,p}(\mathbb R^n)}.$$
Here $\overline{B}$ is the closed unit ball in $\mathbb R^n$.
(ii) Show that $F$ has a compact closure in $X$. Here $X$ is the closure of $\{f \in C_c^\infty(\mathbb R^n\setminus \overline{B}) : f \text{ is radial}\}$ under $L^\infty(\mathbb R^n)$-norm and
$$F = \{f \in C_c^\infty(\mathbb R^n\setminus \overline{B}) : f \text{ is radial and } \|f\|_{W^{1,p}(\mathbb R^n)} \le 1\}.$$

Problem 8 (qiuzhen-qualify 2025-spring)
Section: Analysis

(15 pts) Given a domain $\Omega \subset \mathbb C$ and a point $p \in \Omega$, define
$$c(p) = \sup\{|f'(p)| : f \in (\Omega, D)_p\},$$
where $D$ is the open unit disk $|z| < 1$ and $(\Omega, D)_p$ is the set of holomorphic maps $f : \Omega \to D$ with $f(p) = 0$.
(i) Find the value of $c(p)$ when $\Omega = \mathbb C \setminus \{0,1,2\}$.
(ii) Find the value of $c(p)$ when $\Omega = D$.