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 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} ⁠. All derivatives are zero at 0, i.e. f ′ ( 0 ) = f ″ ( 0 ) = f ‴ ( 0 ) = ⋯ = 0 {\displaystyle f'(0)=f''(0)=f'''(0)=\cdots =0} , and are also all zero at all positive integers.

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 ξ_n 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 + 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 to the Fabius function f: u ( t ) = { f ( t + 1 ) , | t | < 1 0 , | t | ≥ 1 . {\displaystyle u(t)={\begin{cases}f(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}.} It fulfills the delay differential equation^[2] d d t u ( t ) = 2 u ( 2 t + 1 ) − 2 u ( 2 t − 1 ) . {\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).} (See Delay differential equation for 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]

• 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.

log ⁡ f ( x ) = − log 2 ⁡ x 2 log ⁡ 2 + log ⁡ x ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 − ( 1 2 + 1 + log ⁡ log ⁡ 2 log ⁡ 2 ) log ⁡ x − log 2 ⁡ ( − log ⁡ x ) 2 log ⁡ 2 + log ⁡ log ⁡ 2 ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 + ( 6 γ 2 + 12 γ 1 − π 2 − 6 log 2 ⁡ log ⁡ 2 12 log ⁡ 2 − 7 log ⁡ 2 12 − log ⁡ π 2 ) + log 2 ⁡ ( − log ⁡ x ) 2 log ⁡ 2 ⋅ log ⁡ x − log ⁡ log ⁡ 2 ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 ⋅ log ⁡ x + O ( 1 log ⁡ x ) {\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 ⁠ x → 0 + {\displaystyle x\to 0^{+}} ⁠, where γ {\displaystyle \gamma } is Euler's constant, and γ 1 {\displaystyle \gamma _{1}} is the Stieltjes constant. Equivalently,

log ⁡ f ( 2 − n ) = − n 2 log ⁡ 2 2 − n log ⁡ n + ( 1 + log ⁡ 2 2 ) n − log 2 ⁡ n 2 log ⁡ 2 + ( 6 γ 2 + 12 γ 1 − π 2 12 log ⁡ 2 − 7 log ⁡ 2 12 − log ⁡ π 2 ) − log 2 ⁡ n 2 n log 2 ⁡ 2 + O ( 1 n ) {\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 ⁠ n → ∞ {\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. (1979), Non-classical methods of the approximation theory in boundary value problems (in Russian), Kiev: Naukova Dumka

This mathematical analysis–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e