Fabius function

Graph of the Fabius function on the interval [0,1].

Extension of the function to the nonnegative real numbers.

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Template:Harvs.

This function satisfies the initial condition $f(0)=0$, the symmetry condition $f(1-x)=1-f(x)$ for $0\le x\le 1,$ and the functional differential equation

$f^{\prime }(x)=2f(2x)$

for $0\le x\le 1/2.$ It follows that $f(x)$ is monotone increasing for $0\le x\le 1,$ with $f(1/2)=1/2$ and $f(1)=1$ and $f^{\prime }(1-x)=f^{\prime }(x)$ and $f^{\prime }(x)+f^{\prime }(\frac{1}{2}-x)=2.$

It was also written down as the Fourier transform of

$\hat{f}(z)=\prod _{m=1}^{\mathrm{\infty }}{(\cos \frac{\pi z}{2^m})}^m$

by Template:Harvs.

The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of

$\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}{\xi }_n,$

where the Template:Math are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of $\frac{1}{2}$ and a variance of $\frac{1}{36}$.

There is a unique extension of Template:Mvar to the real numbers that satisfies the same differential equation for all x. This extension can be defined by Template:Math for Template:Math, Template:Math for Template:Math, and Template:Math for Template:Math with Template:Mvar a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.

The Rvachev up function is closely related: Template:Math.

Values

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:

• $f(1)=1$
• $f(\frac{1}{2})=\frac{1}{2}$
• $f(\frac{1}{4})=\frac{5}{72}$
• $f(\frac{1}{8})=\frac{1}{288}$
• $f(\frac{1}{16})=\frac{143}{2073600}$
• $f(\frac{1}{32})=\frac{19}{33177600}$
• $f(\frac{1}{64})=\frac{1153}{561842749440}$
• $f(\frac{1}{128})=\frac{583}{179789679820800}$

with the numerators listed in Template:OEIS2C and denominators in Template:OEIS2C.

References

• Template:Citation
• Template:Citation
• Template:Cite thesis
• Template:Cite arXiv
• Template:Cite arXiv (an English translation of the author's paper published in Spanish in 1982)
• Alkauskas, Giedrius (2001), "Dirichlet series associated with Thue-Morse sequence", preprint.
• Rvachev, V. L., Rvachev, V. A., "Non-classical methods of the approximation theory in boundary value problems", Naukova Dumka, Kiev (1979) (in Russian).

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