%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      03/04/2025
%%%%
%%%% Purpose:   Chebyshev Polynomials
%%%%

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\title{Chebyshev Polynomials}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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\institute[Millersville University of Pennsylvania]
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{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
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\the\year}

\subject{Talks}
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\begin{document}

\begin{frame}
  \titlepage
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  % You might wish to add the option [pausesections]
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% Since this a solution template for a generic talk, very little can
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% - Exactly two or three sections (other than the summary).
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% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will:
\begin{itemize}
\item explore the Chebyshev ordinary differential equation,
\item define the Chebyshev polynomials, and
\item describe the properties of the Chebyshev polynomials,
\end{itemize}

\end{frame}
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\begin{frame}\frametitle{Chebyshev Ordinary Differential Equation}

The \textbf{Chebyshev differential
  equation} is
\[
%\label{eq:spc:chebyshev}
(1 - x^{2}) y'' - x y' + \alpha^{2} y
	= 0
\mbox{ for $-1 < x < 1$,}
\]
where $\alpha$ is a constant.
The solutions of this equation have many applications in the physical
sciences and in numerical mathematics.

\vspace*{0.10in}

Two linearly independent solutions to this ODE are
\begin{align*}
y_{1}(x)
	&= 1 + \sum_{n=1}^{\infty} \frac{\prod_{k=0}^{n-1} ((2k)^{2} -
          \alpha^{2})}{(2n)!}	x^{2n}	\\
y_{2}(x)
	&= x + \sum_{n=1}^{\infty} \frac{\prod_{k=0}^{n-1} ((2k+1)^{2}
          - \alpha^{2})}{(2n+1)!} x^{2n+1}	.
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Chebyshev Polynomials}

If $\alpha = 2N$ where
$N$ is a nonnegative integer,
the solution reduces to the polynomial,
\[
y_{1}(x)
	= 1 + \sum_{n=1}^{N} (-1)^{n} \frac{\prod_{k=0}^{n-1} (N + k)
     (N - k)}{(2n)!} (2x)^{2n}	.
\]

\vspace*{0.10in}

If $\alpha = 2N+1$ for
some nonnegative integer $N$, the
solution becomes
the polynomial,
\[
y_{2}(x)
	= x + \frac{1}{2} \sum_{n=1}^{N} (-1)^{n}
     \frac{\prod_{k=0}^{n-1} (N + k + 1) (N - k)}{(2n+1)!} (2x)^{2n+1}	.
\]

\vspace*{0.10in}

By convention these polynomials are denoted $T_{n}(x)$ and are called
the \textbf{Chebyshev polynomials of the first kind}.

\end{frame}
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\begin{frame}\frametitle{Examples}

\begin{center}
\begin{tabular*}{\columnwidth}{@{}rcl@{}}
$T_{0}(x)$	& = & $\displaystyle 1$	\\
$T_{1}(x)$	& = & $\displaystyle x$	\\
$T_{2}(x)$	& = & $\displaystyle 2x^{2} - 1$	\\
$T_{3}(x)$	& = & $\displaystyle 4x^{3} - 3x$	\\
$T_{4}(x)$	& = & $\displaystyle 8x^{4} - 8x^{2} + 1$	\\
$T_{5}(x)$	& = & $\displaystyle 16x^{5} - 20x^{3} + 5x$	\\
\end{tabular*}
\end{center}

\end{frame}
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\begin{frame}\frametitle{Illustration}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{chebyshevps}}
\end{center}

\end{frame}
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\begin{frame}\frametitle{Properties (1 of 3)}

\begin{lemma}
%\label{lma:spc:chebyshevcosine}
For each nonnegative integer $n$,
\[
%\label{eq:spc:chebyshevcosine}
T_{n}(\cos \theta)
	= \cos (n \theta)	.
\]
\end{lemma}
As a consequence, $T_{n}(x) = \cos (n \arccos x)$.

\vspace*{0.10in}

\textbf{Recurrence relation:}
\[
T_{n+1}(x)
	= 2 x \, T_{n}(x) - T_{n-1}(x)
\]

\end{frame}
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\begin{frame}\frametitle{Properties (2 of 3)}

\begin{theorem}
%\label{thm:spc:chebyshevzeros}
The polynomial $T_{n}(x)$ has $n$ distinct zeros in the interval $(-1,
1)$
located at
\[
x_{k}
	= \cos \frac{(2k-1) \pi}{2n}
\mbox{ for $k = 1, 2, \ldots, n$.}
\]
\end{theorem}

\begin{theorem}
%\label{thm:spc:chebyshevleading}
For $n \in \mathbb{N}$ the leading coefficient of $T_{n}(x)$ is $2^{n-1}$.
\end{theorem}

\begin{theorem}
%\label{thm:spc:chebyshevexplicit}
For $n \in \mathbb{N}$ the $n$th Chebyshev polynomial is given by the
formula
\[
%\label{eq:spc:chebyshevexplicit}
T_{n}(x)
	= 2^{n-1} \prod_{k=1}^{n} \left( x - \cos \frac{(2k-1) \pi}{2n} \right)	.
\]
\end{theorem}

\end{frame}
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\begin{frame}\frametitle{Properties (3 of 3)}

\begin{theorem}
%\label{thm:spc:chebyshevorthog}
For $m, \, n \in \{ 0, 1, 2, \ldots \}$ the Chebyshev polynomials
are orthogonal on $[-1, 1]$ with respect to the weighting function
$(1-x^{2})^{-1/2}$ and
\[
%\label{eq:spc:chebyshevorthog}
\int_{-1}^{1} \frac{T_{m}(x) T_{n}(x)}{\sqrt{1-x^{2}}} \, dx
	= \left\{ \begin{array}{rl}
\pi & \mbox{if $m=n=0$,}	\\
\displaystyle \frac{\pi}{2} \delta_{mn} & \mbox{otherwise,}
\end{array} \right.
\]
where $\delta_{mn}$ is the Kronecker delta function.
\end{theorem}

\begin{theorem}
%\label{thm:spc:chebyshevrodrigues}
For $n \in \{ 0, 1, 2, \ldots \}$ the Chebyshev polynomial of degree
$n$, $T_{n}(x)$ can be expressed as
\[
%\label{eq:spc:chebyshevrodrigues}
T_{n}(x)
	= \frac{(-2)^{n} n!}{(2n)!} (1-x^{2})^{\frac{1}{2}}
\frac{d^{n}}{dx^{n}} \left[ (1-x^{2})^{n-\frac{1}{2}} \right]	.
\]
\end{theorem}

\end{frame}
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\begin{frame}\frametitle{Generalized Fourier Series}

If $f(x)$ is sufficiently smooth on $(-1, 1)$ then
\[
f(x)
	= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} T_{n}(x)	,
\]
where
\[
%\label{eq:spc:chebyshevcoeff}
a_{n}
	= \frac{2}{\pi} \int_{-1}^{1} \frac{f(x) T_{n}(x)}{\sqrt{1-x^{2}}} \, dx	,
\]
for $n = 0, 1, 2, \ldots$.

\end{frame}
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%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
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