Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).

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

${\displaystyle f'(x)=2f(2x)}${\displaystyle f'(x)=2f(2x)}

for ${\displaystyle 0\leq x\leq 1/2.}${\displaystyle 0\leq x\leq 1/2.} It follows that ${\displaystyle f(x)}${\displaystyle f(x)} is monotone increasing for ${\displaystyle 0\leq x\leq 1,}${\displaystyle 0\leq x\leq 1,} with ${\displaystyle f(1/2)=1/2}${\displaystyle f(1/2)=1/2} and ${\displaystyle f(1)=1}${\displaystyle f(1)=1} and ${\displaystyle f'(1-x)=f'(x)}${\displaystyle f'(1-x)=f'(x)} and ${\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}${\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}

It was also written down as the Fourier transform of

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

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

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

There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f (x) = 0 for x ≤ 0, f (x + 1) = 1 − f (x) for 0 ≤ x ≤ 1, and f (x + 2^r) = −f (x) for 0 ≤ x ≤ 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.

The Rvachëv up function^[1] is closely related: $${\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\0,円\quad |t|\geq 1\end{cases}}}$${\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\0,円\quad |t|\geq 1\end{cases}}} which fulfills the Delay differential equation ^[2] $${\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).}$${\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).} (see Another example).

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:^{[3] [4]}

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

with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757 .

${\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O\!\left({\frac {1}{\log x}}\right)\end{aligned}}}${\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O\!\left({\frac {1}{\log x}}\right)\end{aligned}}}

for ${\displaystyle x\to 0^{+},}${\displaystyle x\to 0^{+},} where ${\displaystyle \gamma }${\displaystyle \gamma } is Euler's constant, and ${\displaystyle \gamma _{1}}${\displaystyle \gamma _{1}} is the Stieltjes constant. Equivalently,

${\displaystyle \log f\!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)-{\frac {\log ^{2}n}{2n\log ^{2}2}}+O\!\left({\frac {1}{n}}\right)}${\displaystyle \log f\!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)-{\frac {\log ^{2}n}{2n\log ^{2}2}}+O\!\left({\frac {1}{n}}\right)}

for ${\displaystyle n\to \infty .}${\displaystyle n\to \infty .}

• Fabius, J. (1966), "A probabilistic example of a nowhere analytic C ^∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, doi:10.1007/bf00536652, MR 0197656, S2CID 122126180
• 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).
• 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).

• Types of functions
• Mathematical analysis stubs

• Articles with short description
• Short description matches Wikidata
• All stub articles