% ch01slidesa.tex

\begin{slide}
\begin{center}\large\bf
Periodic Functions and Orthogonal Systems
\end{center}

\begin{itemize}
\item {Periodic Functions}
\item {Even and Odd Functions}
\item {Properties of Even and Odd Functions}
\item {Properties of Periodic Functions}
\item {Piecewise-Defined Functions}
\item {Representations of Even and Odd Extensions}
\item {Integration and Differentiation of Piecewise-Defined Functions}
\item {Inner Product}
\item {Orthogonal Functions}
\item {Trigonometric System Details}
\end{itemize}

\end{slide}

\begin{slide}
\header{Periodic Functions}

\bigskip

\BF{Definition}.  A function $f$ is $T$-periodic if and only if $f(t+T)=f(t)$ for all $t$.


\bigskip


\BF{Definition}. The $\floor$ function is defined by
 $$\floor(x) = \mbox{greatest integer not exceeding $x$}.$$

\bigskip


\BF{Theorem}. Every function $g$ defined on $0 \le x \le T$ has a $T$-periodic extension $f$ defined on the whole real line by the formula
$$f(x) = g(x-T\floor(x/T)).$$
\end{slide}



\begin{slide}
\header{Even and Odd Functions}

\bigskip

\BF{Definition}. A function $f(x)$ is said to be \EM{even} provided
$$ f(-x) = f(x),\quad \mbox{for all $x$}. $$
A function $g(x)$ is said to be \EM{odd} provided
$$ g(-x) = -g(x),\quad \mbox{for all $x$}. $$

\medskip

\BF{Definition}. Let $h(x)$ be defined on $[0,T]$.

The \DF{even extension} $f$ of $h$ to $[-T,T]$ is defined by
$$ f(x) = \left\{ \begin{array}{lcl} h(x) & 0 \le x \le T, \\ h(-x) & -T \le x < 0.\end{array}\right.$$

\medskip

Assume $h(0)=0$. The \DF{odd extension} $g$ of $h$ to $[-T,T]$ is defined by
$$ g(x) = \left\{ \begin{array}{lcl} h(x) & 0 \le x \le T, \\ -h(-x) & -T \le x < 0.\end{array}\right.$$

\end{slide}



\begin{slide}
\header{Properties of Even and Odd Functions}

\bigskip

\BF{Theorem}. Even and odd functions have the following properties.
\begin{itemize}
  \item The product and quotient of an even and an odd function is odd.
  \item The product and quotient of two even functions is even.
  \item The product and quotient of two odd functions is even.
  \item Linear combinations of odd functions are odd.
  \item Linear combinations of even functions are even.
\end{itemize}


\bigskip

\BF{Theorem}. Among the trigonometric functions, the cosine and secant are even and the sine and cosecant, tangent and cotangent are odd.
\end{slide}



\begin{slide}
\header{Properties of Periodic Functions}

\medskip

\BF{Theorem}. If $f$ is $T$-periodic and continuous, and $a$ is any real number, then
$$\int_0^T f(x)dx = \int_a^{a+T} f(x)dx.$$

\bigskip


\BF{Theorem}. If $f$ and $g$ are $T$-periodic, then
\begin{itemize}
\item $c_1f(x)+c_2g(x)$ is $T$-periodic for any constants $c_1$, $c_2$
\item $f(x)g(x)$ is $T$-periodic
\item $f(x)/g(x)$ is $T$-periodic
\item $h(f(x))$ is $T$-periodic for any function $h$
\end{itemize}
\end{slide}

\begin{slide}
\header{Piecewise-Defined Functions}

\medskip

\BF{Definition}.
For $a\le b$, define $\dd\pulse(x,a,b) = \left\{ \begin{array}{ll} 1 & a \le x < b, \\ 0 & \mbox{otherwise}. \end{array}\right.
$

% The function $\pulse(x,a,a)$ is called a \DF{point pulse}.

\bigskip

\BF{Definition}. Assume that $a\le x_1 \le x_2 \le \cdots \le x_{n+1}\le b$. Let $f_1$, $f_2$, \ldots, $f_n$ be continuous functions defined on $-\infty < x < \infty$.
A \DF{piecewise continuous function} $f$ on a closed interval $[a,b]$ is a sum
$$ f(x) = \sum_{j=1}^n f_j(x)\pulse(x,x_j,x_{j+1}).$$



\bigskip

 If additionally $f_1$, \ldots, $f_n$ are continuously differentiable on $-\infty < x < \infty$,
 then sum $f$ is called a \DF{piecewise continuously differentiable} function.
\end{slide}

\begin{slide}
\header{Representations of Even and Odd Extensions}

\bigskip

\BF{Theorem}. The following formulas are valid.
\begin{itemize}
\item If $f$ is the even extension on $[-T,T]$ of a function $g$ defined on $[0,T]$, then
$$f(x) = g(x)\pulse(x,0,T)+g(-x)\pulse(x,-T,0).$$
\item If $f$ is the odd extension on $[-T,T]$ of a function $h$ defined on $[0,T]$, then
$$f(x) = h(x)\pulse(x,0,T)-h(-x)\pulse(x,-T,0).$$
\item The $2T$-periodic extension $F$ of $f$ is given by
$$ F(x) = f(x-2T\floor(x/(2T))).$$
\end{itemize}
\end{slide}



\begin{slide}\header{Integration and Differentiation of Piecewise-Defined Functions}

\bigskip
\BF{Theorem}.
Assume the piecewise-defined function is given on $[a,b]$ by the pulse formula
$$ f(x) = \sum_{j=1}^n f_j(x)\pulse(x,x_j,x_{j+1}).$$
Then
$$ \int_a^b f(x)dx = \sum_{j=1}^n \int_{x_j}^{x_{j+1}} f_j(x)dx.$$
If $x$ is not a division point $x_1$, \ldots, $x_{n+1}$, and each $f_j$ is differentiable, then
$$ f'(x) = \sum_{j=1}^n f_j'(x)\pulse(x,x_j,x_{j+1}).$$
\end{slide}

\begin{slide}
\header{Inner Product}

\bigskip

\BF{Definition}. Define the inner product symbol $\innerprod{f,g}$ by the formula
$$\innerprod{f,g}=\int_a^b f(x)g(x)dx.$$
If the interval $[a,b]$ is important, then we write $\innerprod{f,g}_{[a,b]}$.

\medskip

The \DF{inner product} $\innerprod{\cdot,\cdot}$ has the following properties:

\bigskip

\begin{itemize}
\item $\innerprod{f,f} \ge 0$ and for continuous $f$, $\innerprod{f,f}=0$ implies $f=0$.
\item $\innerprod{f,g_1+g_2}=\innerprod{f,g_1}+\innerprod{f,g_2}$
\item $c\innerprod{f,g}=\innerprod{cf,g}$
\item $\innerprod{f,g}=\innerprod{g,f}$
\end{itemize}
\end{slide}

\begin{slide}
\header{Orthogonal Functions}

\bigskip

\BF{Definition}. Two nonzero functions $f$, $g$ defined on $a\le x \le b$ are said to be \DF{orthogonal} provided $\innerprod{f,g}=0$.

\bigskip

\BF{Definition}. Functions $f_1$, \ldots, $f_n$ are called an \DF{orthogonal system} provided
 \begin{itemize}
  \item $\innerprod{f_j,f_j} > 0$ for $j=1,\ldots,n$
  \item $\innerprod{f_i,f_j}=0$ for $i \ne j$
 \end{itemize}


 \bigskip

 \BF{Theorem}. An orthogonal system $f_1$, \ldots, $f_n$ on $[a,b]$ is linearly independent on $[a,b]$.

 \bigskip

 \BF{Theorem}. The first three Legendre polynomials $P_0(x)=1$, $P_1(x)=x$, $P_2(x)=\frac{1}{2}(x^2-1)$ are an orthogonal system on $[-1,1]$. In general, the system $\{ P_j(x)\}_{j=0}^\infty$ is orthogonal on $[-1,1]$.

  \bigskip

 \BF{Theorem}. The trigonometric system $1$, $\cos x$, $\cos 2x$, \ldots, $\sin x$, $\sin 2x$, \ldots is an orthogonal system on $[-\pi,\pi]$.
 \end{slide}

\begin{slide}
 \header{Trigonometric System Details}

 \bigskip

 \BF{Theorem}. The orthogonal trigonometric system $1$, $\cos x$, $\cos 2x$, \ldots, $\sin x$, $\sin 2x$, \ldots on $[-\pi,\pi]$ has the orthogonality relations
 $$
 \begin{array}{lcl}
  \innerprod{\sin nx,\sin mx} &=& \int_{-\pi}^{\pi} \sin nx\sin mx dx =
    \left\{\begin{array}{ll} 0 & n \ne m \\ \pi & n=m \end{array}\right.
    \\[2ex]
    \innerprod{\cos nx,\cos mx} &=& \int_{-\pi}^{\pi} \cos nx\cos mx dx =
    \left\{\begin{array}{ll} 0 & n \ne m \\ \pi & n=m>0 \\ 2\pi & n=m=0 \end{array}\right.
    \\[2ex]
    \innerprod{\sin nx,\cos mx} &=& \int_{-\pi}^{\pi} \sin nx\cos mx dx = 0.
 \end{array}
 $$
\end{slide}
