%&latex % february 14, 2000 \documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amscd} \usepackage[mathscr]{eucal} \newcommand{\Aut}{{\mathrm{Aut} }} \newcommand{\End}{{\mathrm{End} }} \newcommand{\GL}{{\mathbf{GL} }} \newcommand{\Gal}{{\mathrm{Gal}\, }} \newcommand{\G}{{\mathbf G}_{m}} \newcommand{\Hom}{{\mathrm{Hom} }} \newcommand{\Tr}{{\mathrm{Tr}\, }} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z} \newcommand{\R}{\mathbf R} \newcommand{\C}{\mathbf C} \newcommand{\F}[1]{\mathbf{F}_{#1}} \newcommand{\Fbar}[1]{\overline{\mathbf{F}}_{#1}} \newcommand{\fr}[1]{\,{\rm Frob}_{\, #1}} \newcommand{\q}[1]{\mathbf{Q}_{#1}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\msr}[1]{\mathscr{#1}} \newcommand{\mbf}[1]{\mathbf{#1}} \newcommand{\mfr}[1]{\mathfrak{#1}} \newcommand{\mr}[1]{\mathrm{#1}} \newcommand{\ok}[1]{O_{#1}} \newcommand{\re}[1]{{\rm Re} (#1)} \newcommand{\im}[1]{{\rm Im} (#1)} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{pro}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \theoremstyle{definition} \newtheorem{Def}[thm]{Definition} %\theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{exa}[thm]{Example} \begin{document} \title{Lecture 30} \author{Meg Onoda} \address{Louisiana State University \\ Department of Mathematics \\ Baton Rouge, LA 70803, USA} \date{\today} \email{onoda@math.lsu.edu} \maketitle \section{Lecture 30} \thm\label{T:IV.3.1} If $G$ is compact abelian, then the charactristics of $G$ are a complete $L^2$-basis for $G$. This means every $f \in L^2(G,\mu)$ can be written uniquely as\\ \[ f = \sum_{\chi \in \widehat{G}} c_\chi \cdot \chi . \] This $f$ converges in $L^2(\widehat{G}, \widehat{\mu})$ with $\hat{\mu (\chi)} = 1$, and where $c_\chi \in \C$. This is Fourier coefficients.\\ Let's compute the Fouriere coefficients for $ f = \sum_{x \in \chi} c_\chi \cdot \chi$. \[ \begin{array}{lll} (f, \phi )_G &=& \sum_{\chi} c_\chi (\chi, \psi)\\ &=& c_\phi \cdot (\phi, \psi)_G \end{array} \] This implies that $c_\phi = \frac{(f, \phi)_G}{(\phi, \phi)_G}$.\\ Here is some example.\\ Let $G=T$ ($T$ as a unit circle). Then $d\mu = \frac{d \theta}{2\pi}$ and so $\int_T d\mu = 1$. We know that $\widehat{G} = \Z$ and $\chi_n (t) = t^n$, where $t \in T, n \in Z$. If we take polar coordinates, then $\chi_n (\theta) = e^{in\theta}$ and this function is periodic. So, if $f \in L^2 (T, d\theta)$, and $f = \sum_{n \in Z} c_n \cdot e^{in \theta}$, where\\ \[ \begin{array}{lll} c_n &=& \frac{(f, \chi_n)_T}{(\chi_n, \chi_n)_T}\\ &=& \int_T f(t)\overline{\chi_n (t)} d\mu (t), because (\chi_n, \chi_n)_T = 1\\ &=& \frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-in\theta} d\theta. \end{array} \] This is Fourier coefficients and this $f$ is periodic in $\theta \to \theta + 2\pi$.\\ \Def\label{D:IV.3.1} Let $G$ be a locally compact abelian group and $\widehat{G} =$ group of characters. Given a function, $f:G \longrightarrow \C$. We define its Fourier transform, $\widehat{f}:\widehat{G} \longrightarrow \C$ as \[ \widehat{f}(\chi) = \int_G f(x)\chi (x) d\mu (x) = (f,\chi ^-1)_G \] For example,\\ If $G$ is compact, $f=\sum_{\chi\in \widehat{G}} c_\chi \chi, where c_\chi = \frac{(f,\chi)_G}{(\chi, \chi)_G} = \frac{\widehat{f}(\chi^-1)}{(\chi, \chi)_G}$. Hence \[ f = k \sum_ {\chi\in \widehat{G}}\widehat{f}(\chi^-1)\chi. \] for some constant $k$.\\ For some other example,\\ If $G=\R$, then we know that $\widehat{G}\simeq \R$. For $y\in R$, we have $\chi_y : R \longrightarrow T$ defined as $\chi_y (x)=e^{-2\pi ixy}$. Therefore $f:\R \longrightarrow \C$ and Forier Transformation $\widehat{f}:\R \longrightarrow \C$ defined as, $\widehat{f}(y)=\int_R f(x)\chi_y(x) dx = \int_{-\infty}^{+\infty} f(x) e^{-2\pi ixy} dx$.\\ \begin{thm} \label{T:III.3.2} There is a unique Haar measure $\hat{\mu}$ on $\widehat{G}$ called dual to $\mu$ such that\\ \[ f(s)=\int_{\widehat{G}}^{} \hat{f} (\chi) \chi^{-1}(s) d\hat{\mu}(\chi). \] for sufficiently regular functions $f$ on $G$. This is Fourier inversion. \end{thm} For example,\\ For all $f \in C_0 (G,C)=continuous C-valued functions with compact support.$ Moreover, $(f,f)_G = (\hat{f}, \hat{f})_{\widehat{G}}$. Plancharel's Theorem says, if $G$ is compact, then if $\mu (G) = 1, and \hat{\mu} (G) = 1$, then \[ f(s)=\sum_{\chi \in \widehat{G}} C_\chi^{-1} \cdot \chi^{-1} (s), \] where $c_\chi^{-1} = \hat{f} (x).$ And \[ \hat{f} (x) = \int_G f(s) \chi(s) d\mu(s). \] So, if $G=\R$, then Haar measure is $d\mu = \frac{dx}{\sqrt{2\pi}}$. And $\widehat{G} \simeq \R \longrightarrow d\hat{\mu} = \frac{dy}{\sqrt{2\pi}}$, where $y \in \R$ and $\chi_y:\R \longrightarrow T$ defined by $\chi_y (x) = e^{-2\pi ixy}$. Hence \[ f(x) = \frac{1}{\sqrt{2\pi}} \int_\R \hat{f}(y) e^{2\pi ixy}dy \] and \[ \hat{f}(y)= \frac{1}{\sqrt{2\pi}} \int_\R f(x) e^{-2\pi ixy}dx. \] We have a situation when $G$ is locally compact abelian and $\Gamma \subsetneq G$ as discrete closed subgroup such that $G/\Gamma$ is compact.\\ For example,\\ if $\Z \subsetneq \R, \mathfrak{O}_K \hookrightarrow A_K$ where $K \subsetneq A_K,$ and $K^X \hookrightarrow {J_K}^1.$ Then, $\Gamma ^\bot = {\chi \in \widehat{G} : \chi (\Gamma) \equiv 1} \subsetneq \widehat{G}$. And $\Gamma ^\bot \simeq \widehat{G/\Gamma} and so \widehat{G} / \Gamma ^\bot \simeq \widehat{\Gamma}$.\\ \thm \label{T:III.3.3} (Poisson Summation)\\ Choose Haar measure $\mu$ on $G$ so that $\mu(G/ \Gamma ) = 1$, then\\ \[ \sum_{\gamma \in \Gamma} f(\gamma) = \sum_{\hat{\gamma} \in \Gamma ^\bot} \hat{f} (\hat{\gamma}) \] for all sufficiently regular $f$. %\mbox{}\hfill %$\displaystyle n \atopwithdelims() \displaystyle k$ \hfill\mbox{} \bibliographystyle{amsplain} %\providecommand{\bysame}{\leavevmode\hbox %to3em{\hrulefill}\thinspace} %\begin{thebibliography}{10} %\bibitem{sL} %S. Lang, \emph{Algebraic Number Theory}, Second Edition, Graduate %Texts in Mathematics, {\bf 110}, Springer - Verlag, 1994. %\end{thebibliography} \end{document}