Digit function properties

As @Mark Dominus said you won't be able to solve for your $a_k(n)$, but you can find a Fourier series for $f_n$.

First $f_n(x) = f_0(10^{-n}x)$ and $x = \sum 10^n f_0(10^{-n}x)$ so I will only deal with $f_0$.

$f_0(x+10) = f_0(x)$, so let us extend $f_0$ to the negative numbers by $f_0(x-10)=f_0(x)$ so $f_0$ is periodic over all $\mathbb{R}$. For cleanliness let's also define $$f_0(k)=\lim_{\epsilon\to 0}\frac{f_0(k-\epsilon)+f_0(k+\epsilon)}{2}$$ at the round integers $k$, so e.g. $f_0(2.9999\ldots) = f_0(3.0) = 2.5$ and $f_0(29.9999\ldots)=f_0(30.0)=4.5$. Also let $g_0(x) = f_0(x)-4.5$, then $g_0$ is an odd periodic function and has a Fourier sine series. In fact $g_0$ is the difference of two sawtooth waves. It's fairly straightforward to find $$ g_0(x) = -\frac{10}{\pi}\sum_{k=1}^\infty b_k \sin\left(\frac{k\pi x}{5}\right) $$ where $$ b_k = \begin{cases}0 & \mathrm{if}~10\mid k\\ 1/k & \mathrm{otherwise}\end{cases} $$

I don't know of any literature about this function. I realize this hasn't answered your specific questions, but I hope it's of some interest.