\documentclass[oneside,english,spanish,12pt,a4paper]{report}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}




\hyphenation{cons-tan-te} \hyphenation{re-cu-pe-rar}
\hyphenation{fun-cio-nes} \hyphenation{Shannon}
\hyphenation{u-sa-re-mos} \hyphenation{cons-tan-tes}
\hyphenation{EITFG} \hyphenation{que-re-mos}
\hyphenation{bi-blio-gra-fí-a} \hyphenation{pe-rio-di-za-da}
\hyphenation{Mul-tirre-so-lu-ción} \hyphenation{Gramiano}
\hyphenation{di-fe-ren-cia} \hyphenation{mues-tre-o}
\hyphenation{ge-ne-ra-li-za-da} \hyphenation{su-pon-ga-mos}
\hyphenation{dia-go-nal} \hyphenation{ma-ne-ra}
\hyphenation{ob-te-ne-mos} \hyphenation{au-to-va-lo-res}
\hyphenation{co-rres-pon-dien-tes}
%\hyphenation{}
%\hyphenation{}
%\hyphenation{}
%\hyphenation{}
%\hyphenation{}

\pagestyle{myheadings} \pagestyle{plain}
\usepackage[spanish]{babel}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usepackage{mathrsfs}
%\makeatletter
\usepackage{graphicx}
\usepackage{epsfig}

 \newcommand {\theoremstyle} [1] { }
 \newtheorem{thm}{Teorema}[section]
 \theoremstyle{plain}
 \newtheorem{cor}[thm]{Corolario} %%Delete [thm] to re-start numbering
 \theoremstyle{plain}
 \newtheorem{lem}[thm]{Lema} %%Delete [thm] to re-start numbering
 \theoremstyle{plain}
 \newtheorem{prop}[thm]{Proposici\'on } %%Delete [thm] to re-start numbering
 \theoremstyle{definition}
 \newtheorem{defn}[thm]{Definici\'on}
 \theoremstyle{remark}
 \newtheorem{ej}[thm]{Ejemplo}
 \newtheorem{rem}[thm]{Observaci\'on}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\usepackage{amssymb}
% numeros reales
\newcommand{\R}{\mathbb{R}}
% numeros enteros
\newcommand{\Z}{\mathbb{Z}}
%numeros naturales
\newcommand{\n}{\mathbb{N}}
% para la bibliografía
\newcommand{\y}{y}
%span
\newcommand{\gen}{\mbox{gen}}
%senocardinal
\newcommand{\senc}{\mbox{senc}}
%seno\sum_{k} |\widehat{f} (\w+n) | \right)^2
\newcommand{\sen}{\mbox{sen}}
% numeros naturales
\newcommand{\N}{\mathbb{N}}
%varphi
\newcommand{\p}{\varphi}
%omega
\newcommand{\w}{\omega}
%epsilon
\newcommand{\e}{\varepsilon}
%alpha
\newcommand{\f}{\alpha}

\newcommand{\ind}{\tiny \begin{array}{c} l \in L \\
                 k \in \Z^d \end{array}}


\newenvironment{proof}[1][{\small Demostraci\'on}]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em} \vspace{.08in}}

\newtheorem{thm*}{Teorema}[section]

%\renewcommand\labelitemi{\(\bullet\)}
\makeatother

\begin{document}


\begin{center}
\textbf{\huge{Teor\'ia Geom\'etrica de la Medida}}
\end{center}

\begin{center}
 \bf{\LARGE{Ejercicios}}
\end{center}

\begin{description}

\item[Ejercicio 26:] (Clase 14, 05/06)

Sea $A\in \Z^{d\times d}$ expansiva, $D=\{k_1,\ldots, k_q\}$ un
conjunto de d\'igitos del cociente $\Z^d/A\Z^d$ y $Q$ el conjunto
invariante asociado a $\phi=\{\p_1,\ldots,\p_q\}$, donde
$\p_i(x)=A^{-1}(x+k_i)$.

Si $F=\bigcup_{\gamma\in\Z^d}(Q+\gamma)$, probar que existe $c>0$ tal que para todo $x\in\R^d,\,\, \exists
\,\,x_0\in F$ con $\|x_0-x\|\leq c$.

\item[Ejercicio 27:] (Clase 14, 05/06)

Sea $A\in \Z^{d\times d}$ expansiva y $Q\subseteq\R^d$ compacto, con las siguientes propiedades:
\begin{description}
\item[i)] $|Q\cap Q+\gamma|=0$, si $\gamma\neq 0$,
\item[ii)] Existe un conjunto de d\'igitos $\{k_1,\ldots,k_q\}$ tal que $AQ=\bigcup_{i=1}^{q}Q+k_i$,
\item[iii)] $\bigcup_{\gamma\in\Z^d}(Q+\gamma)=\R^d$.
\end{description}
Si se define la sucesi\'on de subconjuntos
$\{V_j\}_{j\in\Z}\subseteq L^2(\R^d)$ como
$$V_0=\overline{gen\{\chi_{Q+\gamma}:\gamma\in\Z^d\}}\quad\textrm{y}\quad V_j=D_A^jV_0,$$
donde $D_A(f)=|A|^{j/2}f(A^j.-)$,
probar que
\begin{description}
\item [({\it a})] $\overline{\bigcup_{j\in\Z}V_j}=L^2(\R^d).$
\item [({\it b})] $\bigcap_{j\in\Z}V_j=\{0\}$.
\end{description}

\item[Ejercicio 28:] (Clase 14, 05/06)

Sea $A\in \Z^{d\times d}$ expansiva, $D=\{k_1,\ldots, k_q\}$ un
conjunto de d\'igitos del cociente $\Z^d/A\Z^d$ y $Q$ el conjunto
invariante asociado a $\phi=\{\p_1,\ldots,\p_q\}$, donde
$\p_i(x)=A^{-1}(x+k_i)$. Supongamos que $|Q|=1$ y sean
$\psi^1,\ldots,\psi^{q-1}$ funciones de la forma
$\psi^s=\sum_{k_i\in D}\alpha^s_i\chi_{Q+k_i}$ que verifican las
siguientes propiedades:
\begin{description}
\item[i)] $\psi^i\bot\psi^j$, si $i\neq j$,
\item[ii)] $\| \psi^s\|_2=1$ si $1\leq s\leq  q-1$,
\item[iii)] $\int_{\R^d}\psi^j(x)dx=0$, para todo $1\leq j\leq  q-1$.
\end{description}

Probar que $\{q^{j/2}\psi^s(A^jx-\gamma): j\in\Z, \gamma\in\Z^d, s=1,\ldots,q-1\}$ es
una base ortonormal de $L^2(\R^d)$.



\end{description}
\end{document}
