Question on pathological sine function

Question

Some years ago I came across what was defined as "pathological" function defined as: $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^2}\cdot \sin\left(k!\cdot x\right) $$ It was mentioned (in an article I cannot remember) as something that could not be completely drawn because the partial sums become increasingly "ripply" when adding new terms.

I did some experimenting with plotting software and this seems the case, but I don't know if sums of this type are very trivial to build or this is a more special case.

Is this series related to any well known special function ? Has anyone more information on the property of it ?

Thanks in advance

Prospero

Answer 1

A Fourier representation comes to mind. You can imagine writing

$$f(x) = \sum_{j=1}^{\infty} a_j \sin{j x} $$

where

$$ a_j = \begin{cases} 1/k^2 & j=k! \\ 0 & \mathrm{otherwise} \end{cases}$$

for each $k \in \mathbb{N}$. What function $f(x)$ has such a coefficient I cannot say.