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 Jaap Fabius (1966). It was also written down as the Fourier transform of

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

by Børge Jessen and Aurel Wintner (1935).

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

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

where the $ξ n$ are independent uniformly distributed random variables on the unit interval.

This function satisfies the initial condition $f(0)=0$ , the symmetry condition $f(1-x)=1-f(x)$ for $0\leq x\leq 1,$ and the functional differential equation $f'(x)=2f(2x)$ for $0\leq x\leq 1/2.$ It follows that $f(x)$ is monotone increasing for $0\leq x\leq 1,$ with $f(1/2)=1/2$ and $f(1)=1.$ There is a unique extension of $f$ to the real numbers that satisfies the same equation. This extension can be defined by $f (x) = 0$ for $x \leq 0$ , $f (x + 1) = 1 - f (x)$ for $0 \leq x \leq 1$ , and $f (x + 2 r) = - f (x)$ for $0 \leq x \leq 2 r$ with $r$ a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.

Values

The Fabius function is constant zero for all negative arguments, and takes rational values at non-negative dyadic rationals. Those values are given by the following formula:

f\!\left({\frac {n}{2^{m}}}\right)=\sum _{\lambda =0}^{\left\lfloor m/2\right\rfloor }\!\sum _{\mu =0}^{\lambda }\sum _{\nu =1}^{n}\sum _{\kappa =1}^{2^{\mu }}{\frac {(-1)^{\lambda +s_{2}(n-\nu )+s_{2}(\kappa -1)}}{2^{{\binom {m+1}{2}}-{\binom {\mu }{2}}+2\lambda \,(\mu +1)}}}\cdot {\frac {(2\nu -1)^{m-2\lambda }}{(m-2\lambda )!}}\cdot {\frac {(2\kappa -1)^{2\lambda +\mu +1}}{(2\lambda +\mu +1)!}}\!\left(\prod _{\sigma =1}^{\mu }{\frac {1}{4^{\sigma }-1}}\!\right)\!\!\left(\prod _{\sigma =1}^{\lambda -\mu }{\frac {1}{4^{\sigma }-1}}\!\right)\!,

where $s_{2}(n)$ is the sum of digits of n in base-2. Note that $(-1)^{s_{2}(n)}=f(2n+1)=t_{n}$ is just the signed Thue–Morse sequence, satisfying the recurrence $t_{0}=1,\,\,t_{n}=(-1)^{n}\,t_{\lfloor n/2\rfloor }.$

References

Fabius, J. (1966), "A probabilistic example of a nowhere analytic $C \infty$ -function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, doi:10.1007/bf00536652, MR 0197656
Jessen, Børge; Wintner, Aurel (1935), "Distribution functions and the Riemann zeta function", Trans. Amer. Math. Soc., 38: 48–88, doi:10.1090/S0002-9947-1935-1501802-5, MR 1501802
Dimitrov, Youri (2006). Polynomially-divided solutions of bipartite self-differential functional equations (Thesis).
Haugland, Jan Kristian (2016). "Evaluating the Fabius function". arXiv:1609.07999 [math.GM].
Arias de Reyna, Juan (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT].
Arias de Reyna, Juan (2017). "An infinitely differentiable function with compact support: Definition and properties". arXiv:1702.05442 [math.CA]. (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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