- Polynomials are not the only functions that can be used to approximate the known function.
- Another means for representing known functions are approximations that use sines and cosines, called Fourier series.
- Any function can be represented by an infinite sum of sine and cosine terms with the proper coefficients, (possibly with an infinite number of terms).
- Any function,
, is periodic of period
if it has the same value for any two x-values, that differ by
, or
Figure 1:
Plot of a periodic function of period P.
|
Figure 1 shows such a periodic function. Observe that the period can be started at any point on the
-axis.
and
are periodic of period
and
are periodic of period
and
are periodic of period
- We now discuss how to find the
s and
s in a Fourier series of the form
![\begin{displaymath}
f(x)\approx \frac{A_0}{2}+ \sum_{n=1}^{\infty} [A_ncos(nx)+B_nsin(nx)]
\end{displaymath}](img22.png) |
(1) |
The determination of the coefficients of a Fourier series (when a given function,
, can be so represented) is based on the property of orthogonality for sines and cosines. For integer values of
:
 |
(2) |
 |
(3) |
 |
(4) |
 |
(5) |
 |
(6) |
It is related to the same term used for orthogonal (perpendicular) vectors whose dot product is zero. Many
functions, besides sines and cosines, are orthogonal (such as the Chebyshev polynomials).
- To begin, we assume that
is periodic of period
and can be represented as in Eq. 1. We find the values of
and
in Eq. 1 in the following way;
- For
; multiply both sides of Eq. 1 by
, and integrate term by term between the limits of
and
.
Because of Eqs. 2 and 3, every term on the right vanishes except the first, giving
Hence,
is found and it is equal to twice the average value of
over one period.
- For
; multiply both sides of Eq. 1 by
, where
is any positive integer, and integrate:
Because of Eqs. 3,4 and 6 the only nonzero term on the right is when
in the first summation, so we get a formula for the
s;
- For
; multiply both sides of Eq. 1 by
, where
is any positive integer, and integrate:
Because of Eqs. 2, 4 and 5, the only nonzero term on the right is when
in the second summation, so we get a formula for the
s;
It is obvious that getting the coefficients of Fourier series involves many integrations.
Subsections
Cem Ozdogan
2010-12-29