Does an exponential series expansion exist? I know we can expand a function $f:\mathbb{R} \to \mathbb{R}$ with a power series
$$
f(x) = \sum_{n=0}^\infty a_n x^n,
$$
where we can find find $\{a_n\}$ given $f$ using its derivatives about $0$. What I would like to know is whether an "exponential series" exists for $f$. That is, is there a sequence $\{b_n\}$ such that
$$
f(x) = \sum_{n=0}^\infty b_n e^{x n},
$$
and if so, how would we find $b_n$ given a general $f$? Are there conditions on $f$ for which no sequence $\{b_n\}$ can exist?
Edit: To be clear, $x$ here is real, not imaginary, so this is not a Fourier series, and the $e^{xn}$ functions are not an orthonormal basis. Obviously some trivial series do exist, such as $f(x) = e^x$ for $b_n = \delta_{n1}$, but what about more interesting functions such as $f(x) = e^{x^2}$?
 A: Such an expansion exists whenever $f$ is periodic. That is, say $f(x + 2\pi) = f(x)$ for every $x$. Obviously $2\pi$ is just a convention here. By rescaling, you can make it be whatever you want.
Then the Fourier Series associated to $f$ is the sum
$$f(x) = \sum_{-\infty}^\infty a_n e^{2 \pi i n x}$$
Under relatively general conditions (that are still technical enough for me to not want to mention) such a series exists, but periodicity is the important one. We can see periodicity is necessary since each of the $e^{2 \pi i n x}$ are periodic.
In much the same way that a taylor series lets you write $f$ in terms of its derivatives, a fourier series lets your write $f$ in terms of its "harmonics". Importantly, we still have a nice formula for the coefficients:
$$a_n = \int_{-\infty}^\infty e^{-2 \pi i n x} \cdot f ~ \mathrm{d}x$$
In many contexts, these $a_n$ are called $\hat{f}(n)$, and you'll see that notation scattered through the literature. This is a field that is very deep and very broad, so it should be exciting to get some exposure to it!
You can find a relatively polite treatment of the subject in these notes by Tom Leinster

I hope this helps ^_^
