Is Fourier Expansion possible only for Periodic Functions?

Question

I understand from other related posts here that Fourier Expansion is possible only for Periodic Functions.

Also, the text book which I am reading states: "Any Periodic function $f(x)$ of period $2 \cdot \pi$, which satisfies certain conditions known as Dirichlet's Conditions, can be expressed in the form of the series $$a_0 + \sum_{n}(a_n\cos(nx) + b_n\sin(nx))$$, for all values of x in any interval c to c + $2\pi$, of length $2\pi$. The expansion of $f(x)$ in the form of the above series is called Fourier Series".

The text book is full of examples like 'Find Fourier Series for $f(x) = x^2$ or $f(x) = e^{-x}$ ' etc. So are these functions ($x^2$ or $e^{-x})$, periodic?

Answer 1

No. They are usually, but not necessarily, considered on the finite interval, $[-\pi,\pi)$. You can always extend them periodically to $\mathbb{R}$ then by setting $$f(x+2n\pi):= f(x), n\in \mathbb{Z}\backslash\{0\}, x\in [-\pi,\pi)$$

(And yes, the classical Fourier expansion is valid for periodic functions only).

Answer 2

The Fourier Transform is in some sense an attempt to do what you describe. To extend Fourier series to arbitrary functions. However those functions need to obey some restrictions in order to be possible to be Fourier Transformable. There exist a very rich theory on which functions are transformable and which properties the transform has. Very many books on Fourier Transforms or Harmonic Analysis as it is also called as it is a very popular tool in science and engineering. An answer on this site can only help point in the right direction as any real treatment would be too involving to fit a single answer.