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 f ( 0 ) = 0 {\displaystyle f(0)=0} , the symmetry condition f ( 1 − x ) = 1 − f ( x ) {\displaystyle f(1-x)=1-f(x)} for 0 ≤ x ≤ 1 , {\displaystyle 0\leq x\leq 1,} and the functional differential equation
f ′ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)}for 0 ≤ x ≤ 1 / 2. {\displaystyle 0\leq x\leq 1/2.} It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for 0 ≤ x ≤ 1 , {\displaystyle 0\leq x\leq 1,} with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1 {\displaystyle f(1)=1} and f ′ ( 1 − x ) = f ′ ( x ) {\displaystyle f'(1-x)=f'(x)} and f ′ ( x ) + f ′ ( 1 2 − x ) = 2. {\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}
It was also written down as the Fourier transform of
f ^ ( z ) = ∏ m = 1 ∞ ( cos π z 2 m ) 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
∑ n = 1 ∞ 2 − n ξ n , {\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}where the ξn are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of 1 2 {\displaystyle {\tfrac {1}{2}}} and a variance of 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 + 2r) = −f (x) for 0 ≤ x ≤ 2r 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 Rvachev up function is closely related: up(x) = F(1 - |x|) for |x| ≤ 1.
Values
The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:12
- f ( 1 ) = 1 {\displaystyle f(1)=1}
- f ( 1 2 ) = 1 2 {\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}
- f ( 1 4 ) = 5 72 {\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}
- f ( 1 8 ) = 1 288 {\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}
- f ( 1 16 ) = 143 2073600 {\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}
- f ( 1 32 ) = 19 33177600 {\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}
- f ( 1 64 ) = 1153 561842749440 {\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}
- f ( 1 128 ) = 583 179789679820800 {\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}
with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757.
- 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).
References
Sloane, N. J. A. (ed.). "Sequence A272755 (Numerators of the Fabius function F(1/2^n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩
Sloane, N. J. A. (ed.). "Sequence A272757 (Denominators of the Fabius function F(1/2^n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩