Analysis General Exam, January 2009

Problem 1

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a bounded real function.
(a) Fix $\delta>0$, and for every $\alpha \in \mathbb{R}$ consider the set

$$U_{\alpha}:=\{x \in \mathbb{R}: \sup \{f(y): y \in \mathbb{R},|y-x|<\delta\}>\alpha\} .$$

Show that for every $\alpha \in \mathbb{R}$ the set $U_{\alpha}$ is open.
(b) Let $u: \mathbb{R} \rightarrow \mathbb{R}$ be given by

$$u(x):=\lim _{\delta \rightarrow 0}(\sup \{f(y): y \in \mathbb{R},|y-x|<\delta\})$$

Show that the function $u$ is well-defined and Lebesgue measurable.
Similarly, define

$$l(x):=\lim _{\delta \rightarrow 0}(\inf \{f(y): y \in \mathbb{R},|y-x|<\delta\})$$

which, by the same arguments, will also be well-defined and Lebesgue measurable.
(c) Show that the set of points where the function $f$ is continuous is measurable.

Problem 2

Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded, Lebesgue measurable function such that

$$\int_{[0,1]} f(x) x^{k} d x=\frac{1}{k+4}, \quad \text { for } \quad k=0,1,2, \ldots$$

Show that $f(x)=x^{3}$ a.e. on $[0,1]$.

Problem 3

Let $F$ be a Lebesgue measurable subset of $[0,2] \times[0,2]$. For each $x \in[0,2]$ set $F^{x}:=\{y \in[0,2]:(x, y) \in F\}$ and for each $y \in[0,2]$ let $F_{y}:=\{x \in[0,2]:(x, y) \in F\}$. Show that if $\left|F^{x}\right| \leq \frac{1}{8}$ then

$$\left|\left\{y \in[0,2]:\left|F_{y}\right| \geq 1\right\}\right| \leq \frac{1}{4}$$

Here, if $E$ is a Lebesgue measurable subset of $\mathbb{R}$, we denote by $|E|$ its Lebesgue measure.

Problem 4

Let $\left\{f_{n}\right\}_{n \in \mathbb{N}}$ be a sequence of functions such that, for each $n \in \mathbb{N}$ we have $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ and $f_{n} \in L^{2}(\mathbb{R}, d x)$. Assume that for every $m, n \in \mathbb{N}$ such that $m \geq n$ one has that

$$\left\|f_{n}-f_{m}\right\|_{2}<2^{-n}$$

where generically, $\|g\|_{2}$ denotes the $L^{2}$-norm of the function $g \in L^{2}(\mathbb{R}, d x)$.
(a) Show that the sequence $\left\{f_{n}\right\}_{n \in \mathbb{N}}$ converges point-wise a.e. on $\mathbb{R}$.
(b) Let $\left\{u_{n}\right\}_{n \in \mathbb{N}}$ be an orthonormal set of functions, also in $L^{2}(\mathbb{R}, d x)$. Does the series

$$\sum_{n=1}^{\infty} \frac{u_{n}(x)}{2^{n}}, \quad x \in \mathbb{R}$$

converge point-wise a.e. on $\mathbb{R}$ ?
(c) If $x \in \mathbb{R}$ is a point where the series from (1) converges we denote by $F(x)$ its value. Show that $F \in L^{2}(\mathbb{R}, d x)$ and compute its norm $\|F\|_{2}$ explicitly.

Problem 5

(a) Compute the integral

$$I(r)=\int_{0}^{\infty} \frac{d x}{1+x^{r}}, \quad \text { where } \quad r>1$$

Hint: Use contour integral techniques. More specifically, consider a contour integral over $[0, \infty) \cup\left\{z \in \mathbb{C}: z=t e^{2 \pi i / r}, t \in[0, \infty)\right\}$.
(b) Compute

$$\lim _{r \rightarrow \infty} I(r)$$

either directly by examining the limit of the integral above, or by using your answer to part (a).

Problem 6

Let $D_{+}:=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$ be the right half plane and assume that $f: \overline{D_{+}} \rightarrow \mathbb{C}$ is analytic in $D_{+}$, continuous on the closure $\bar{D}_{+}$of $D_{+}$, and satisfies

$$|f(z)| \leq\left\{\begin{array}{lll} 1 & \text { if } & \operatorname{Re} z=0 \\ \ln |z| & \text { if } & |z| \geq 3 \end{array}\right.$$

Show that in fact

$$|f(z)| \leq 1 \quad \text { in } \quad D_{+}$$

Hint: Consider the function $g_{\varepsilon}: \overline{D_{+}} \rightarrow \mathbb{C}$ given by $g_{\varepsilon}(z)=\frac{f(z)}{(1+z)^{\varepsilon}}$ for $\varepsilon>0$.

Problem 7

Let $\left\{f_{n}\right\}_{n \in \mathbb{N}}$ be a bounded sequence of analytic functions in the unit disk $D:=\{z \in \mathbb{C}:|z|<1\}$.
(a) Let $0<r<1$. Show that there exist a subsequence $\left\{f_{n_{j}}\right\}_{j \in \mathbb{N}}$ of the above sequence which converges

$$\lim _{j \rightarrow \infty} f_{n_{j}}(z)=g_{r}(z), \quad z \in D_{r}$$

where $D_{r}:=\{z \in \mathbb{C}:|z|<r\}$ and $g_{r}: D_{r} \rightarrow \mathbb{C}$ is an analytic function.
(b) Does there exist a subsequence of the given sequence $\left\{f_{n}\right\}$ which converges point-wise to a function $g(z)$ analytic on $D$ ?