Fabius function

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

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

:$f'(x) = 2 f(2 x)$

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 $f'(1-x)=f'(x)$ and $f'(x)+f'(\tfrac12-x)=2.$

It was also written down as the Fourier transform of

:$\hat(z) = \prod_^\infty \left(\cos\frac\right)^m$

by .

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

:$\sum_^\infty2^\xi_n,$

where the are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of $\tfrac$ and a variance of $\tfrac$.

There is a unique extension of to the real numbers that satisfies the same differential equation for all ''x''. This extension can be defined by for , for , and for with 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 ''Rvachëv up function'' is closely related: $u(t)=\begin F(t+1),\quad , t, <1 \\ 0, \quad , t, \geq 1 \end$ which fulfills the Delay differential equation
$\fracu(t)=2u(2t+1)-2u(2t-1).$ (see Another example).

Values

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

* $f(1)=1$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$
* $f(\tfrac1) =\tfrac$

with the numerators listed in and denominators in .

References

*
*
*
*
* (an English translation of the author's paper published in Spanish in 1982)
Types of functions
* 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).

{{mathanalysis-stub