Let us consider Takagi-function defined by

$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,

where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.

$T(x)$ has its period $1$, so I am trying to develop its Fourier expansion

$S(T)(x) \colon\!=\sum_{n= -\infty}^{\infty}\left(a_n\,sin(2 \pi n x) + b_n\,cos(2 \pi n x)\right)$.

## Q: Where do we have $T(x) = S(T)(x)$?

For the case of Weierstrass function $W(x) \colon\!= \sum_{n=0}^{\infty}a^n cos(b^n \pi x)$ with $ab > 1 + 3\pi/2$, $ 0 < a < 1$ and $b$ an odd integer, it is already expanded in Fourier series with period $2$ by shape.

By Carlson, we know $T(x) = S(T)(x)$ except for sets of Lebesgue measure $0$. I would like to know whether $T(x) = S(T)(x)$ holds everywhere for Takagi-function as well.