• idea: to express complicated functions by using simple functions

for example:

power series :

• Besides power series, we do have other ways, such as Fourier series

Advantages:

1. easy to differentiate and integrate
2. each term contain only one characteristic frequency
3. To solve PDE and ODE (forbid some numerical methods)
4. can be used to describe non continuous (non differetiable) functions

• A very famous algorithm, FFT Math7270 十大算法

Q: Under what kind of conditions $f(x)$ will have fourier series expansion? A: Dirichlet conditions

1. the function must be periodic: $f(x+p)=f(x)$
2. must be single-valued and continuous except possibly at a finite number of finite discontinuities
3. must have only a limit number of maximum and minimum within 1 period
4. the integrated over 1 period of $|f(x)|$ must converge

• If Dirichlet conditions are satisfied, the fourier series converge to $f(x)$ at all point where f(x) is continuous.
• roughly, why $f(x)$ can be expressed by sine and cosine?

 

 

For a general function $f(x)$,

fourier series of $f(x)$

where $a_r$, $a_0$, $b_r$ are constants called Fourier coefficients

L: period of $f(x)$

• MAIN TASK: to determine the Fourier coefficients of $f(x)$ when its fourier series exists.
• remark: All the terms of a Fourier series are mutually orthogonal. (Trigonometric Identities)

Q: If Fourier Series of $f(x)$ exists, What are coefficients? A: with period L,

$x_0$ can be arbitrary input on $\mathbb{R}$.

Why?

3. when $p=0$:

   when $p\neq 0$:
4. Hence,

   Let $\omega=\frac{2\pi}{T}$ be angular frequency

Example: square-wave function

1. First of all, $f(t)$ is a odd function so we only need to solve the sine part in F.S.

Symmetric properties

1. Odd functions has no cosine terms (all $a_r=0$)
2. Even functions has no cosine terms (all $b_r=0$)

Discontinuous Function

If $f(x)$ is discontinuous, at $x_d$, then Fourier Series converges to

However, Fourier Series will overshoot its value although more terms would reduce the overshoot, it never disappear even in the limit. (Gibbs’ Phenomenon)

Size of overshoot his proportioned to the magnitude of the discontinuity.

Example: square wave

 

Non-periodic function

to expand(延拓为奇函数或偶函数)

Example: Find F.S. for $f(x)=x^2, 0<x<2$

1. L=4

3. For $a_0$, by formula

4. So

• Remark: $\int u d v = u v - \int v d u$

Integration and differentiation

Fourier Series of a function id obtained by integration and differentiation of another one

Example: $f(x)=x^3, 0\leq x\leq 2$.

 

1. By integration,

   and we have

   F.S. don’t have $4x$!

2. to deal with $4x$, by differentiation,

Complex Fourier Series

So we have

with coefficient

$C _ { r } = \frac { 1 } { L } \int _ { x _ { 0 } } ^ { x _ { 0 } + L } f ( x ) \exp ( - \frac { 2 \pi i x } { L } ) d x$

Q: Why we have $C_r$ like?

1. orthogonallity:

2. prove $C_r$

• The relation between $C_r$ and $a_r$ and $b_r$:

Why?

Example: $f(x)=x, -2<x<2$

1. when $r\neq 0$:

   Hint:

2. when $r=0$:

Parsval’s thm.

Remark: $\left|f(x)\right|^2=\left<f(x)|f(x)\right>=\int f(x) g(x) dx$

• P:

1. Consider $f(x)$ and $g(x)$ with period $L$;

Last Updated on 3 years by Yichen Liu