Suppose we have a function $f$ defined over $[a,b]$ to the real numbers, i.e. $f: [a, b] \to \mathbb R.$.
We can approximate this function as Fourier Series.
Suppose $a_n, b_n$ is the Fourier series coefficients,
Then
$$a_n = \frac{2}{b-a} \int_a^bf(x) \cos(nx)$$
$$b_n = \frac{2}{b-a}\int_a^bf(x)\sin(nx) $$
$$f(x) \approx \frac{a_0}{2}+ \sum^\infty_{n=1} a_n\cos(nx) + b_n\sin(nx)$$
My question is, what is the advantage of using the approximate Fourier Series,
rather than the function $f(x)$ itself? When would we use the approximation over the funtion itself?
Best Answer
Often a problem can be made easier by reducing it to individual frequencies. I've provided an example below that occurs when working with linear differential equations.
Consider the oscillator equation with a time dependent periodic forcing function,
$$ \frac{d^2 x}{dt^2} + k x = F(t)$$
This can be a very difficult problem for an arbitrary forcing function $F$, but if we decompose $F$ into its fourier series then we really just need to solve three simple problems.
$$ \frac{d^2 x}{dt^2} + k x = a_n\cos(n\omega t)$$
$$ \frac{d^2 x}{dt^2} + k x = b_n \sin(n\omega t)$$
$$ \frac{d^2 x}{dt^2} + k x = a_0$$
All three of the above equations are easily solvable. Lets call the various solutions to these equations $x_\lambda(t)$. We can construct the solution to the original problem by just taking linear combinations of these solutions,
$$ x(t) = \sum_\lambda x_\lambda(t) $$