Nowhere analytic, infinitely differentiable 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 ).
This function satisfies the initial condition
f
(
0
)
=
0
{\displaystyle f(0)=0}
f
(
1
−
x
)
=
1
−
f
(
x
)
{\displaystyle f(1-x)=1-f(x)}
0
≤
x
≤
1
,
{\displaystyle 0\leq x\leq 1,}
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.}
f
(
x
)
{\displaystyle f(x)}
0
≤
x
≤
1
,
{\displaystyle 0\leq x\leq 1,}
f
(
1
/
2
)
=
1
/
2
{\displaystyle f(1/2)=1/2}
f
(
1
)
=
1
{\displaystyle f(1)=1}
f
′
(
1
−
x
)
=
f
′
(
x
)
{\displaystyle f'(1-x)=f'(x)}
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 ξ n independent uniformly distributed random variables on the unit interval . That distribution has an expectation of
1
2
{\displaystyle {\tfrac {1}{2}}}
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 ) = 0x ≤ 0f (x + 1) = 1 − f (x )0 ≤ x ≤ 1 , and f (x + 2r f (x )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 Rvachëv up function [ 1]
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}}}
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).}
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:[ 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 OEIS : A272757
Asymptotic
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^{+},}
γ
{\displaystyle \gamma }
Euler's constant , and
γ
1
{\displaystyle \gamma _{1}}
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 .}
References
Fabius, J. (1966), "A probabilistic example of a nowhere analytic C ∞ 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 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 ]. 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).
