Fourier series.

where and are real numbers, is called 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 .

The Fourier series of is given by:

- .
- For , .
- For , .

Noah Dana-Picard 2007-12-28