Problem 1 (qiuzhen-qualify 2024-spring)
Section: Analysis

Compute the Fourier series

$$\sum_{n\in\mathbb{Z}} a_n e^{inx}$$

of the function $f(x)=\cos(\alpha x)$ defined on $[-\pi,\pi]$ where $\alpha$ is not an integer, and use it to compute

$$\sum_{n=1}^{\infty} \frac{1}{9n^2-1}.$$

Problem 2 (qiuzhen-qualify 2024-spring)
Section: Analysis

For any non-negative integer $n\in\mathbb{Z}_+$, compute

$$\int_{0}^{\pi/2} \left(\frac{\sin(nx)}{\sin x}\right)^2 \, dx.$$

Problem 3 (qiuzhen-qualify 2024-spring)
Section: Analysis

Determine the stability (asymptotically stable/stable but not asymptotically stable/unstable) of the zero solution for the second order differential equation

$$\frac{d^2 y}{dt^2} + \left(\frac{dy}{dt}\right)^3 + y = 0.$$

Problem 4 (qiuzhen-qualify 2024-spring)
Section: Analysis

Given $s>0$, prove that the following limit $u$ exists in the sense of distribution (in $\mathcal{D}'(\mathbb{R})$) and compute its Fourier transform $\hat u(\xi)$:

$$u = \lim_{\varepsilon\to0^+} \frac{1}{x^2 - s^2 + i\varepsilon}.$$

(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 5 (qiuzhen-qualify 2024-spring)
Section: Analysis

Given a sequence of functions $f_n\in L^1(\mathbb{R})$ such that $f_n$ converges to $f\in L^1(\mathbb{R})$ almost everywhere.

(i) Show that $f_n$ may not converge to $f$ in $L^1$ norm by providing a counterexample.

(ii) If in addition $\|f_n\|_{L^1}$ converges to $\|f\|_{L^1}$, then $f_n$ converges to $f$ in $L^1$ norm.

Problem 6 (qiuzhen-qualify 2024-spring)
Section: Analysis

Let $X$ be a reflexive complex Banach space, $C\subset X$ be a closed convex subset. Show that for all $x_0\in X$, there exists $y_0\in C$ such that

$$\|x_0 - y_0\| = \inf_{y\in C} \|x_0 - y\|.$$

Problem 7 (qiuzhen-qualify 2024-spring)
Section: Analysis

Let $B_r$ denote the ball $B(0,r)$ in $\mathbb{R}^2$ centered at the origin with radius $r>0$. Let $\phi(x)=\max\{|x|-1,0\}$, for $x\in\mathbb{R}^2$, be the distance function from the unit ball $B_1$.

(i) Let $1\le p<\infty$. Prove that there is a finite number $C>0$ so that

$$\|u\|_{L^p(B_3\setminus \overline{B_1})} \le C\,\|\phi\,\nabla u\|_{L^p(B_3\setminus \overline{B_1})} + C\,\|u\|_{L^p(B_3\setminus \overline{B_2})}$$

for all $u\in C^1(B_3\setminus \overline{B_1})$ as long as the right hand side is finite.

(ii) Let $1\le p<2$ and $p^* = 2p/(2-p)$. Prove that there is a finite number $C>0$ so that

$$\|\phi\,u\|_{L^{p^*}(B_3\setminus \overline{B_1})} \le C\,\|\phi\,\nabla u\|_{L^p(B_3\setminus \overline{B_1})} + C\,\|u\|_{L^p(B_3\setminus \overline{B_2})}$$

for all $u\in C^1(B_3\setminus \overline{B_1})$ as long as the right hand side is finite.

Problem 8 (qiuzhen-qualify 2024-spring)
Section: Analysis

Let $\phi: \mathbb{R}^2\to\mathbb{R}$ be a Schwartz function. Consider the Laplace's equation in the half space as follows,

$$\begin{cases}
\Delta_{x,y,z}\,u(x,y,z)=0, & (x,y,z)\in \mathbb{R}^3_+ := \mathbb{R}^2\times[0,\infty),\\
u\big|_{z=0} = \phi, & \\
\displaystyle \lim_{z\to\infty} \sup_{x,y\in\mathbb{R}} |u(x,y,z)| = 0. &
\end{cases}$$

(i) Prove that, for any non-negative integer $n\in\mathbb{Z}_+$, there exists an absolute constant $C$, which is independent of $n$ and $\phi$, such that the following estimate holds,

$$\sum_{|\alpha|\le n} \int_{S_n} |\nabla^{\alpha}_{x,y} u(x,y,z)|^2\,dx\,dy\,dz \le C\left( \sum_{|\alpha|\le n} \int_{\mathbb{R}^2} |\nabla^{\alpha}_{x,y} \phi|^2\,dx\,dy \right).$$

(ii) Prove that there exists a constant $C(\phi)$, which depends on $\phi$, such that the following estimate holds,

$$\forall z\in[0,\infty),\quad \sup_{x,y\in\mathbb{R}} |u(x,y,z)| \le C(\phi)\,(1+|z|)^{-1}.$$