\documentclass{article}

\newcommand{\T}{\ensuremath{\mathbf{T}}}
\newcommand{\R}{\ensuremath{\mathbf{R}}}

\title{Syllabus for E. Curry's Oral Qualifying Exam,\\ version 2}
\date{\begin{abstract} The primary topic for my exam is harmonic analysis
and wavelets.  The secondary topic is probability theory.  References
for each topic are given at the end of each section.  Supplementary
references are given where applicable. \end{abstract}}

\begin{document}
\maketitle

Topic I: Harmonic Analysis, Wavelets, Methods, and Related Material

\begin{enumerate}
\item definitions of Fourier coefficients, Fourier series (over \T)
and transform (over \R)
\item summability methods: summability kernels and Fourier multipliers
 \begin{enumerate}
 \item partial sums, Dirichlet "kernel" and corresponding multipliers
 \item Fejer's kernel $\sigma_{n}$ and corresponding multipliers
 \item Poisson kernel $P_{r}$ and corresponding multipliers
 \item Gauss' kernel $G_\lambda$ (for transforms over \R)
 \item norm convergence of summability methods ($L^1$, general $L^p$)
 \item pointwise convergence of summability methods
 \end{enumerate}
\item uniqueness theorem for Fourier coefficients
\item decay of Fourier coefficients
 \begin{enumerate}
 \item Riemann-Lebesgue Lemma
 \item bound from the $L^1$ norm of a function
 \item bound from the $L^1$ norm of derivatives if a function is differentiable
 \item bound from the variance, if a function is of bounded variation
 \item bound from the modulus of continuity
 \item bound if a function is in $\mbox{Lip}_{\alpha}$
 \item theorem that if a sequence $\{a_n\}$ is such that $\sum |a_n|^2
= \infty$, then there is no condition on the moduli of the terms that
can ensure that $\{a_n\}$ is the sequence of Fourier coefficients on an
$L^1(\T)$ function
 \end{enumerate}
\item effects of rotation, translation, and dilation of Fourier coefficients
\item $L^2$ theory of Fourier series and transforms
 \begin{enumerate}
 \item Bessel's Inequality
 \item Riesz-Fischer Theorem
 \item Parseval's formula
 \item Plancherel's formula
 \end{enumerate}
\item conditions for existence of Fourier Inversion Formula
\item $L^p$ theory ($1<p<\infty$) of Fourier series and transforms
 \begin{enumerate}
 \item Marcinkiewicz Interpolation Theorem
 \item convergence of partial sums in $L^p$ norm
 \item Hilbert transform, principal value
 \item Hausdorff-Young Theorem
 \item additional reference: Stein and Weiss. Introduction to Fourier
Analysis on Euclidean Spaces. Princeton University Press, Princeton,
NJ; 1971.
 \end{enumerate}
\item Poisson summation formula
\item Hardy-Littlewood maximal function and related methods
 \begin{enumerate}
 \item Vitali Covering Lemma
 \item maximal function is weak type on $L^1$, bounded on $L^p$ ($1<p$)
 \item Calderon-Zygmund decomposition
 \end{enumerate}
\item basics of MRA wavelets (\T and \R)
 \begin{enumerate}
 \item wavelets, scaling functions
 \item multiresolution analyses
 \item Riesz sequences, Riesz Lemma
 \item construction of MRA wavelets
 \item periodic MRA wavelets
 \item compactly supported MRA wavelets
 \item smooth MRA wavelets
 \end{enumerate}
\item specific wavelets
 \begin{enumerate}
 \item Haar wavelets
 \item Meyer wavelets
 \item spline wavelets
 \item Daubechies D3 wavelet
 \item construction of Daubechies D3 wavelet
 \end{enumerate}
\item some function spaces
 \begin{enumerate}
 \item $\mbox{BMO}_p$, $\mbox{BMO}$
 \item $H_1^q$, $H_1$
 \item relation between the two
 \item $q$-atoms
 \end{enumerate}
\item unconditional convergence
 \begin{enumerate}
 \item of series
 \item unconditional bases ($L^1$, general $L^p$)
 \item Khintchine's inequality
 \end{enumerate}
\item Uncertainty Principle
 \begin{enumerate}
 \item Heisenberg Inequality (for $f \in L^2$, $a$, $b$ real, $\int (x-a)^2
|f|^2$, $\int (\xi-b)^2 |\hat{f}|^2$ $> \frac{||f||^4_{L^2}}{16\pi^2}$)
 \item If $f \in L^1$, $|supp(f)||supp(\hat{f})|<\infty$, then $f=0$.
 \item $P_A$, $Q_B$ projection operators in time and frequency, resp., $A$,
$B$ both of finite measure, then (i) $P_A Q_B$ is a bounded operator
from $L^p$ to $L^q$, (ii) $f \neq 0 \in L^2$
$\epsilon-\mbox{concentrated}$ on $A$, $\hat{f}$
$\delta-\mbox{concentrated}$ on $B$, then $1-\epsilon-\delta \leq
\mbox{ operator norm of } P_A Q_B$
 \item primary reference: Folland and Sitaram. The Uncertainty Principle: A
Mathematical Survey.  J Fourier Analysis and Applications. \bf{3}, 3; 1997.
 \end{enumerate}
\item primary references (unless noted otherwise):
 \begin{enumerate}
 \item Katznelson. An Introduction to Harmonic Analysis.  Dover
Publications, Inc., New York; 1976.
 \item Krantz. A Panorama of Harmonic Analysis. Carus Mathematical
Monographs 27.  MAA; 1999.
 \item Wojtaszczyk. A Mathematical Introduction to Wavelets. London
Mathematical Society Student Texts 37. Cambridge University Press;
1997.
 \end{enumerate}
\end{enumerate}

\pagebreak
Topic II: Probability

\begin{enumerate}
\item basics
 \begin{enumerate}
 \item probability spaces
 \item random variables
 \item distribution functions
 \item expectation
 \item variance
 \item moment generating functions
 \item independence
 \item convergence (in probability, almost sure, in distribution)
 \end{enumerate}
\item properties
 \begin{enumerate}
 \item Tchebyshev Inequality
 \item Chernov Inequality
 \item Levy Reflection Principle
 \item Borel-Cantelli Lemma
 \end{enumerate}
\item tail stuff
 \begin{enumerate}
 \item tail $\sigma-\mbox{algebras}$
 \item Kolmogorov's zero-one law
 \end{enumerate}
\item convergence theorems
 \begin{enumerate}
 \item weak law of large numbers
 \item strong law of large numbers
 \item Kolmogorov's Three-Series Theorem
 \end{enumerate}
\item conditional expectation, Radon-Nikodym Theorem
\item properties of conditional expectation
 \begin{enumerate}
 \item if $X$ is independent of the $\sigma-\mbox{algebra}$ conditioned on
 \item tower property
 \item product of random variables, where one is measurable wrt the $\sigma-\mbox{algebra}$ being conditioned on
 \item basic measure theory properties (positivity, monotone convergence, dominated convergence, Fatou's Lemma, Jensen's Inequality)
 \end{enumerate}
\item martingales
\item optional stopping times
\item Optional Stopping Theorem
\item Bernoulli random walks
\item two theorems about stopped Bernoulli random walks
 \begin{enumerate}
 \item $\mathbf{P}(\bigcap_{-\infty}^{\infty} {T_k < \infty}) = 1$
 \item $\mathbf{P}(T_b < \infty) = (p/q)^b < 1$
 \end{enumerate}
\item primary reference: lecture notes from Professor Ocone's class on
Probability Theory, Rutgers University, Fall 2000.
\end{enumerate}

\end{document}




