# Fourier series.

Definition 10.8.1   A series of the form

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

Example 10.8.2   is a trigonometric series.

The series of coefficients is . By Thm  prop series limit comparison, it is convergent. Thus the given trigonometric series is absolutely convergent.

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

. Recall that the infinite family of functions

is an orthogonal system on the segment .

Definition 10.8.3   Let be an integrable function on the segment . The Fourier coefficients of with respect to the orthogonal system are given by:

The Fourier series of is given by:

Example 10.8.4   Let . Then we have:
• .
• For , .
• For , .

Noah Dana-Picard 2007-12-28