\documentclass[12pt]{article}
\usepackage[width=16cm,height=22cm]{geometry}
\usepackage[intlimits]{amsmath}
\usepackage{amsfonts,amssymb,amsthm,mathtools}
\usepackage{ifthen}

\theoremstyle{definition}
\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{objectives}{Objectives}
\newtheorem*{requirements}{Requirements}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\NaturalNumbers}{{\mathbb{N}}}
\newcommand{\RealNumbers}{{\mathbb{R}}}
\newcommand{\ExtendedRealNumbers}{{\overline{\RealNumbers}}}
\newcommand{\PositiveRealNumbers}{{{\RealNumbers}_+}}
\newcommand{\ComplexNumbers}{{\mathbb{C}}}
\newcommand{\eps}{\varepsilon}
\newcommand{\hsp}[1]{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}
\newcommand{\ubrace}[1]{\underbrace{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}_?}
\newcommand{\bigubrace}[1]{\underbrace{\mbox{}\vphantom{\sum_0^0}\hspace{#1}\mbox{}}_?}
\newcommand{\vertstrut}{{\displaystyle\vphantom{\int}}}
\newcommand{\bigvertstrut}{{\displaystyle\vphantom{\int_0^1}}}
\newcommand{\Range}[1]{{\mathcal{R}(#1)}}
\newcommand{\ClosedRange}[1]{{\mathcal{CR}(#1)}}
\DeclareMathOperator{\closure}{clos}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
\DeclareMathOperator{\spectrum}{Sp}
\DeclareMathOperator{\pointspectrum}{Sp_{\mathit{p}}}
\DeclareMathOperator{\enumber}{e}
\DeclareMathOperator{\imagunit}{i}

\makeatletter
\newcommand{\mytitle}[2][empty]{
\newcommand{\titlename}{#2}
\newcommand{\shorttitlename}{\ifthenelse{\equal{#1}{empty}}{#2}{#1}}
\renewcommand{\@oddfoot}{\hfill{}\shorttitlename{},\quad{}page~{\thepage}~of~\pageref{endlabel}\hfill}
\begin{center}\bfseries\LARGE\titlename\end{center}\par\medskip}
\makeatother

\begin{document}

\mytitle[Analytic functions of two variables vanishing on the conjugate diagonal]%
{Analytic functions of two variables\\
vanishing on the conjugate diagonal}

\medskip\noindent
This text is a rough draft.

\medskip\noindent
These exercises are based on the article of Karel Stroethoff,
The Berezin transform and operators on spaces of analytic functions.

\begin{objectives}
Let $\Omega$ be an open subset of the complex plane $\ComplexNumbers$.
Denote by $\Omega^\ast$ the set
\[
\Omega^\ast\eqdef\bigl\{w\in\ComplexNumbers\colon\ \overline{w}\in\Omega\bigr\}.
\]
Suppose that $h\colon\Omega\times\Omega^\ast\to\ComplexNumbers$ is analytic
and $h(z,\bar z)=0$ for all $z\in\Omega$.
Prove that $h=0$.
This result is used by Karel Stroethoff to prove
that the Berezin transform of bounded linear operators is an injective map.
\end{objectives}

\begin{requirements}
Analytic functions of real variable,
analytic functions of complex variable,
analytic functions of two complex variables,
linear independent functions.
\end{requirements}

\clearpage
\section*{Linear independency of the exponent functions}

\begin{exercise}
Recall the definition of the \emph{Vandermonde matrix}
generated by some numbers $a_1,\ldots,a_n\in\ComplexNumbers$
and the formula for its determinant.
\par\medskip
\[
V(a_1,\ldots,a_n)\eqdef\hsp{10em}
\]
\par\medskip
\[
\det V(a_1,\ldots,a_n)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Denote by $C^k(\RealNumbers,\ComplexNumbers)$ the set
of the $k$-times continuously differentiable functions $\RealNumbers\to\ComplexNumbers$.
\end{exercise}

\begin{exercise}
Let $f_1,\ldots,f_n\in C^{n-1}(\RealNumbers,\ComplexNumbers)$ and $x\in\RealNumbers$.
Recall the definition of the \emph{Wronskian} of $f_1,\ldots,f_n$ at the point $x$:
\par\medskip
\[
W(f_1,\ldots,f_n)(x)\eqdef\hsp{6em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $f_1,\ldots,f_n\in C^{n-1}(\RealNumbers,\ComplexNumbers)$ be linearly dependent.
Prove that for all $x\in\RealNumbers$
\[
W(f_1,\ldots,f_n)(x)=0.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $f_1,\ldots,f_n\in C^{n-1}(\RealNumbers,\ComplexNumbers)$
and $x_0\in\RealNumbers$ such that
\[
W(f_1,\ldots,f_n)(x_0)\ne0.
\]
Prove that $f_1,\ldots,f_n$ are linealy independent.
\end{exercise}

\bigskip

\begin{exercise}
Let $\alpha_1,\ldots,\alpha_n\in\ComplexNumbers$ be pairwise different.
Define the functions $f_j\colon\RealNumbers\to\ComplexNumbers$, $j\in\{1,\ldots,n\}$, by
\[
f_j(x)\eqdef \enumber^{\alpha_j x}.
\]
Prove that $f_1,\ldots,f_n$ are linearly independent.
\end{exercise}


\clearpage
\section*{Homogeneous polynomials of two variables\\
vanishing on the conjugate diagonal}

\bigskip
In this section we consider a $m$-homogeneous polynomial of two complex variables,
that is, a function $P\colon\ComplexNumbers^2\to\ComplexNumbers$ of the form
\[
P(z,w)\eqdef\sum_{k=0}^m a_k z^k w^{m-k}.
\]

\bigskip

\begin{exercise}
Let $r>0$.
Suppose that $P(z,\bar z)=0$ for all $z\in\ComplexNumbers$ with $|z|=r$.
Prove that for all $\theta\in\RealNumbers$
\[
\sum_{k=0}^m a_k \enumber^{2\imagunit k\theta}=0.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $r>0$ and $P(z,\bar z)=0$ for all $z\in\ComplexNumbers$ with $|z|=r$.
Prove that
\[
a_0=\ldots=a_m=0.
\]
\end{exercise}


\clearpage
\section*{General case}

\bigskip

\begin{exercise}
Let $z_0$ be an interior point of $\Omega$
and $r>0$ be such a radius that
\[
\bigl\{z\in\ComplexNumbers\colon\ |z-z_0|<r\bigr\}\subseteq\Omega.
\]
Explain why $h$ can be written as
\[
h(z,w)=\sum_{m=0}^\infty P_m(z-z_0,w-w_0),
\]
where for every $m\in\{0,1,2,\ldots\}$
the function $P_m$ is homogeneous polynomial of two variables of degree $m$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $z\in\ComplexNumbers$ with $|z|<r$.
Define $g\colon(-1,1)\to\ComplexNumbers$ by
\[
g(t)\eqdef h(z_0+tz,z_0+t\bar z).
\]
Calculate $g(t)$ and prove that $g$ is analytic.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Suppose that $h(z,\bar z)=0$ for all $z\in\Omega$.
Prove that $P_m(z,\bar z)=0$ for all $z\in\ComplexNumbers$ with $|z|<r$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Suppose that $h(z,\bar z)=0$ for all $z\in\Omega$.
Prove that $h=0$.
\end{exercise}


\label{endlabel}
\end{document}
