A series of the form
are real numbers, is called a trigonometric series
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.
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
be an integrable function on the segment
. The Fourier coefficients
with respect to the orthogonal system
are given by:
The Fourier series
is given by:
. Then we have:
The graphs of the first terms of a Fourier series.