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\begin{document}

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\begin{center}
 \LARGE 
 \bf
 The Dirichlet kernel
\end{center}

\

\begin{center}
   \begin{tabular}{c}
   \includegraphics[width=435pt, keepaspectratio=true]{./images/dirich.eps}
  \end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 Spiral $\alpha=1/2$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp01}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 ``Spiral'' $\alpha=1$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp02}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 ``Spiral'' $\alpha=65/64$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp03}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 Joined spiral $\alpha=1/2$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp01j}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 Joined spiral $\alpha=1$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp02j}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 Joined spiral $\alpha=65/64$
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=420pt, keepaspectratio=true]{./images/im_examp03j}
\end{tabular}
\end{center}

\newpage



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\begin{center}
 \LARGE 
 \bf
 Branches for $\alpha=1$ ($t_0=4\cdot 10^2+10$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/branch11}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 Branches for $\alpha=1$ ($t_0=12^2-5\cdot 12+2$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/branch12}
\end{tabular}
\end{center}

\newpage



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\begin{center}
 \LARGE 
 \bf
 Branches for $\alpha=1$ ($t_0=4\cdot 11^2-11$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/branch13}
\end{tabular}
\end{center}

\newpage



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\begin{center}
 \LARGE 
 \bf
 A branch for $\alpha=65/64$ ($t_0=7$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/brano1}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 A branch for $\alpha=13/10$ ($t_0=7$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/brano2}
\end{tabular}
\end{center}

\newpage


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\begin{center}
 \LARGE 
 \bf
 A branch for $\alpha=\sqrt{5}$ ($t_0=7$)
\end{center}

\

\begin{center}
\begin{tabular}{c}
 \includegraphics[height=430pt, keepaspectratio=true]{./images/brano3}
\end{tabular}
\end{center}

\newpage












In {\small\url{http://matematicas.uam.es/~fernando.chamizo/dark/images/zzz.avi}} there is an animation with the plot of the trigonometric sum for $0.05\le \alpha\le 3$.

\end{document}
