## 11. Fourier series expansion.

In the previous section we have seen that a function *φ*_{n}
of the vector space Φ_{n} can be expressed by

The functions *φ*_{n} have the following properties:

- their integral mean value in [-π;π] is 0;
- they are continue and differentiable in [-π;π].

If *n* is a finite number, Φ_{n} doesn't include all the possible
functions with these properties.
We can nevertheless conjecture that the more *n* increases, the more the set
Φ_{n} approaches the set Φ of all the periodic functions *φ(x)*,
with period 2π, continue, differentiable and having null integral mean value in [-π;π],
that is

with

this conjecture can be generalized and extended to the set P of all the real periodic
functions *p(x)*, with period 2π, continue, differentiable and having integral
mean value
so that

This expression, found by
J. Fourier,
is commonly said the
Fourier series expansion and has been theorically validated by many mathematicians
of the 19th and 20th centuries, which have extended its applicability also to less
restrictive conditions.

This series expansion can be easily adapted to periodic functions with whatever period 2L
by replacing the *x* variable with the *ξ* of the proportion
.

If a function *p(x)* is odd, all the coefficients *a*_{k} are 0 and
one needs to explicitly calculate only the coefficients *b*_{k}.

The following JavaScript application allows you to observe that the more the number of harmonics, that is, the number of terms in the summation, increases,
the more the sum approximates the function *p(x) = x*. It, like the following ones, works only if your browser allows popups.

The abscissas *x*_{A} and *x*_{B} are expressed as multiple of *π*.

If a function *p(x)* is even, all the coefficients *b*_{k} are 0 and
one needs to explicitly calculate only the coefficients *a*_{k}.

The following JavaScript application allows you to observe that the more the number of harmonics, that is, the number of terms in the summation, increases,
the more the sum approximates the function *p(x) = x*^{2}.

The abscissas *x*_{A} and *x*_{B} are expressed as multiple of *π*.

The expansion of a polynomial is obtained by adding the expansions of its monomials.

Example.

The following JavaScript application allows you to observe that the more the number of harmonics, that is, the number of terms in the summation, increases,
the more the sum approximates the function *p(x) = x*^{2}-2x.

.