Review of Fourier Analysis

From last semesters class we know how to compute the Fourier series of a function $f$ defined on the interval $(-\pi,\pi)$, namely

$$C_n=\frac1{2\pi}\int_{-\pi}^{\pi} f(t)e^{-int}\,dt$$

and

$$S_f(t)=\sum_{n=-\infty}^{\infty} C_ne^{int}.$$

Fourier series will be the most important tool in this project. In order to use this tool we need to review some of their properties. This will be done by doing some examples. Which you should complete in this work book.

1. Let $f_1,f_2,\dots,f_N$ be a finite sequence of $N$ functions with converging Fourier series $S_{f_1},S_{f_2},\dots,S_{f_N}$. Prove that
   $$\overline{f}=\frac1N\left(f_1+f_2+\cdots+f_N\right)$$
   has a converging Fourier series $S_{\overline{f}}$ given by
   $$S_{\overline{f}}=\frac1N\left(S_{f_1}+S_{f_2}+\cdots+S_{f_N}\right)$$

2. Suppose that $f$ and $g$ are two continuous functions, such that $\vert f-g\vert<\epsilon$ for some $\epsilon>0$. Let $C_n(f)$ and $C_n(g)$ be the Fourier coefficients for these two functions. Prove, that there exists a constant $C>0$ such that
   $$\sqrt{\sum_{n=-\infty}^{\infty}\vert C_n(f)-C_n(g)\vert^2}<C\epsilon.$$

3. Suppose that $f$ and $g$ are two continuous functions, such that $\vert f-g\vert<\epsilon$ for some $\epsilon>0$. Let $C_n(f)$ and $C_n(g)$ be the Fourier coefficients for these two functions. Prove, that there exists a constant $C>0$ such that
   $$\vert C_n(f)-C_n(g)\vert<C\epsilon$$
   for all $n$.

If functions are defined on a different interval, say $[-L,L]$, we may compute the Fourier series again using some obvious modifications.

1. Write down the equations for the Fourier coefficients and the Fourier series in this situation.
2. Restate the statements of number 2 and number 3 above in this changed situation.

In order to compare two sound signals, we need to be able to measure their difference. This leads to the concept of norms.

Definition 1   Let $V$ be a vector space. A norm $\Vert\cdot\Vert$ is a mapping from $V$ to the non-negative real numbers which satisfies

1. $\Vert v\Vert=0$, if and only if $v=0$.
2. $\Vert\alpha v\Vert=\vert\alpha\vert\Vert v\Vert$ for any $v\in V$ and any $\alpha\in\mathbb{C}$.
3. $\Vert u+v\Vert\le \Vert u\Vert+\Vert v\Vert$ for all $u,v\in V$.

To become more familiar with this concept you need to complete the following exercises:

1. Let $V=\mathbb{C}$. Show that the absolute value satisfies the requirements of a norm.
2. Let $V$ be the set of all continuous functions on $[-\pi,\pi]$. For $f\in V$, let
   $$\Vert f\Vert=\max\{\vert f(t)\vert:t\in [-\pi,\pi]\}$$
   Show that this is a norm.
3. Let $V$ be the set of all continuous functions on $[-\pi,\pi]$. For $f\in V$, let
   $$\Vert f\Vert=\max\{\vert C_n(f)\vert\}$$
   Show that this is a norm.
4. Let $V$ be the set of all continuous functions on $[-\pi,\pi]$. For $f\in V$, let
   $$\Vert f\Vert=\left(\sum_{n=-\infty}^{\infty} \vert C_n\vert^2\right)^{1/2}$$
   Show that this is a norm.