GENERAL EXAM - ANALYSIS

August, 2010
Closed book, closed notes. Please pledge. In each problem, justify all assertions, show calculations, and identify those theorems which you invoke in your arguments.

Problem 1

Let $f(x)$ be a real, normalized function in $L^{2}(\mathbf{R})$,

$$\int_{\mathbf{R}}|f(x)|^{2} d x=1$$

Show that ( $m$ is Lebesgue measure):
(a) For $c>0$,

$$m\{x:|f(x)| \geq c\} \leq \frac{1}{c^{2}}$$

(b) For $c>0$ and $r>-1 / 2$,

$$\left|\int_{[0, c]} x^{r} f(x) d x\right| \leq \frac{c^{\frac{2 r+1}{2}}}{\sqrt{2 r+1}}$$

(c) Suppose that additionally,

$$\int_{\mathbf{R}} x^{2}|f(x)|^{2} d x<\infty$$

Show that then

$$\int_{\mathbf{R}}|f(x)| d x<\infty$$

Problem 2

Let $0<r<1$ and let

$$\begin{aligned} S_{N}(x) & =\sum_{n \geq 1}^{N} \frac{\cos (n x)}{n^{1+r}} \\ S(x) & =\sum_{n \geq 1}^{\infty} \frac{\cos (n x)}{n^{1+r}} \end{aligned}$$

(a) Show that there is a constant $c_{1}$ such that

$$\left|S_{N}(x)-S_{N}(y)\right| \leq c_{1}|x-y| N^{1-r}$$

(b) Show that there is another constant $c_{2}$ such that

$$\left|S(x)-S_{N}(x)\right| \leq c_{2} N^{-r}$$

(c) By suitable choice of $N$ depending on $x, y$, show that $S(x)$ is Hölder continuous with index $r$, i.e., there is a finite $c_{3}$ such that

$$|S(x)-S(y)| \leq c_{3}|x-y|^{r}$$

Problem 3

Let $(X, \mu)$ be a measure space, and $\operatorname{let}\left\{f_{n}\right\}$ be a sequence of real-valued integrable functions on $X$ which converges to the function $f$ in an $L^{1}$-sense. Suppose that moreover,

$$\sum_{n} \int_{X}\left|f_{n}(x)-f(x)\right| d \mu(x)<\infty$$

Show that $\left\{f_{n}\right\}$ converges pointwise a.e. to $f$. To do so, consider

$$\mu\left(\cup_{m \geq N}\left\{x:\left|f_{m}(x)-f(x)\right| \geq \epsilon\right\}\right)$$

Problem 4

Let $(X, \mu)$ be a finite measure space, and let $\left\{f_{n}\right\}$ be a sequence of real-valued measurable functions converging pointwise a.e. to a measurable function $f$. We say that the sequence $\left\{f_{n}\right\}$ has uniformly absolutely continuous integrals if for every $\epsilon>0$ there exists $\delta>0$ such that

$$\int_{E}\left|f_{n}\right| d \mu<\epsilon$$

for all $n$ whenever $E$ is a measurable set with $\mu(E)<\delta$. Show that in this case, $f_{n} \rightarrow f$ in the norm of $L^{1}(\mu)$.

Problem 5

Show that for every $\epsilon>0$, the function

$$f(z)=\sin z+\frac{1}{z-i}$$

has infinitely many zeros in the set $\{z:|\operatorname{Im} z|<\epsilon\}$.

Problem 6

For $n$ an even integer greater than or equal to 4 , compute

$$\int_{-\infty}^{\infty} \frac{x^{2}}{x^{n}+1} d x$$

Show all estimates carefully. Your answer should be a "clearly real" number.

Problem 7

Suppose $f$ is meromorphic (analytic except for poles) in $\mathbb{C}$. Show that if

$$\int_{\gamma}[p(z)]^{2} f(z) d z=0$$

for every polynomial $p(z)$ and every piecewise smooth closed curve $\gamma$ not passing through a pole of $f$, then $f$ is entire. Hint: First show $f$ is entire under the assumption $\int_{\gamma} p(z) f(z) d z=0$ for all such $p$ and $\gamma$.

Problem 8

Suppose $f$ is entire and $|f(z)| \leq|z|^{2}$ for all $z$. Find all possibilities for $f$.