Yesterday I learned about the strange Fabius function $f$ in this question. Given my interest in neural networks and the fact that this function has a distinct sigmoid shape, I became curious about how to calculate this function.
Three formulas are given on the wikipedia page above:
First regarding it's Fourier Transform: $$ \mathcal F(f(x))(z)=\hat f (z) = \prod_{m=1}^{\infty} \left(\cos\left(\frac{\pi z}{2^m}\right)\right)^m$$
A probabilistic formula regarding it says that it equals the cumulative distribution function of
$$\sum_{n=1}^{\infty}2^{-n}\xi_n, \text{ where } \xi_n = U(0,1)$$
- A functional equation is also given, which it fulfils:
$$f'(x) = 2f(2x)$$
So now to the question ... given these equations, or what we can derive from them, what would be a practical approach to calculate function values on an equally spaced grid for this function?
In other words to estimate $$f(\Delta_t n), \,\, \text{ where }\,\,\cases{ n= \{0,\cdots ,2^{m}-1\}\\ \Delta_t = 2^{-m}}$$
Bonus points for considering all in all computational resources required for the calculations.
This article answers your question: "Evaluating the Fabius function, Jan Kristian Haugland". https://arxiv.org/pdf/1609.07999.pdf