Fourier series.

Definition 10.8.1   A series of the form

$\displaystyle \frac 12 a_0 + \underset{n=1}{\overset{+ \infty}{\sum}} (a_n \cos nx + b_n \sin nx)$    

where $ a_n$ and $ b_n$ are real numbers, is called a trigonometric series.

Example 10.8.2   $ \underset{n=1}{\overset{+ \infty}{\sum}} \frac {\cos nx}{n^2}$ is a trigonometric series.

The series of coefficients is $ \underset{n=1}{\overset{+ \infty}{\sum}} \frac {1}{n^2}$ . By Thm  prop series limit comparison, it is convergent. Thus the given trigonometric series is absolutely convergent.

Let $ L$ be a positive real number. In Linear Algebra, we saw that on the vector space of integrable functions on the segment $ [-L,L]$ , an inner product is defined by

$\displaystyle < f,g > = \frac {1}{L}\int_{-L}^L f(t)g(t)dt$    

. Recall that the infinite family of functions

$\displaystyle \left\{ 1 , \cos \frac {n \pi x}{L}, \sin \frac {n \pi x}{L} \; ; \; n=1,2,... \right\}$    

is an orthogonal system on the segment $ [-L,L]$ .

Definition 10.8.3   Let $ f$ be an integrable function on the segment $ [-\pi, \pi ]$ . The Fourier coefficients of $ f$ with respect to the orthogonal system $ \left\{ 1 , \cos \frac {n \pi x}{L}, \sin \frac {n \pi x}{L} \; ; \; n=1,2,... \right\}$ are given by:

$\displaystyle a_0$ $\displaystyle = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) dx$    
$\displaystyle a_n$ $\displaystyle = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \cos nx dx \; , \; n=1,2,...$    
$\displaystyle b_n$ $\displaystyle = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \sin nx dx \; , \; n=1,2,...$    

The Fourier series of $ f$ is given by:

$\displaystyle f(x) \approx \frac 12 a_0 + \underset{n=1}{\overset{+ \infty}{\sum}} (a_n \cos nx + b_n \sin nx).$    

Example 10.8.4   Let $ f(x)= \begin{cases}0 , -\pi \leq x < 0  1 , 0 \leq x < \pi \end{cases}$ . Then we have:

Figure 7: The graphs of the first terms of a Fourier series.
\begin{figure}\mbox{\epsfig{file=fourier.eps,height=8cm}}\end{figure}
Noah Dana-Picard 2007-12-28