Fabius function

{{short description|Nowhere analytic, infinitely differentiable function}}

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by {{harvs|last=Fabius|first=Jaap|year=1966|txt}}.

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

: $f'(x) = 2 f(2 x)$

for $0 \le x \le 1/2.$ It follows that $f(x)$ is monotone increasing for $0 \le x \le 1,$ with $f(1/2)=1/2$ and $f(1)=1$ and $f'(1-x)=f'(x)$ and $f'(x)+f'(\tfrac12-x)=2.$

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 {{harvs|last1=Jessen|first1=Børge|and|last2=Wintner|first2=Aurel|year=1935|txt}}.

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

: $\sum_{n=1}^\infty2^{-n}\xi_n,$

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

There is a unique extension of {{mvar|f}} to the real numbers that satisfies the same differential equation for all x. This extension can be defined by {{math|1=f{{hsp}}(x) = 0}} for {{math|x ≤ 0}}, {{math|1=f{{hsp}}(x + 1) = 1 − f{{hsp}}(x)}} for {{math|0 ≤ x ≤ 1}}, and {{math|1=f{{hsp}}(x + 2^r) = −f{{hsp}}(x)}} for {{math|0 ≤ x ≤ 2^r}} with {{mvar|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{{cite web | url=https://oeis.org/A288163 | title=A288163 - Oeis }} is closely related: $u(t)=\begin{cases} F(t+1),\quad |t|<1 \\ 0, \quad |t|\geq 1 \end{cases}$ which fulfills the Delay differential equation{{cite arXiv | eprint=1702.06487 | author1=Juan Arias de Reyna | title=Arithmetic of the Fabius function | year=2017 | class=math.NT }}

$\frac{d}{dt}u(t)=2u(2t+1)-2u(2t-1).$ (see 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:{{Cite OEIS |A272755 |Numerators of the Fabius function F(1/2^n). }}{{Cite OEIS |A272757 |Denominators of the Fabius function F(1/2^n). }}

$f(1)=1$
$f(\tfrac1{2}) =\tfrac{1}{2}$
$f(\tfrac1{4}) =\tfrac{5}{72}$
$f(\tfrac1{8}) =\tfrac{1}{288}$
$f(\tfrac1{16}) =\tfrac{143}{2073600}$
$f(\tfrac1{32}) =\tfrac{19}{33177600}$
$f(\tfrac1{64}) =\tfrac{1153}{561842749440}$
$f(\tfrac1{128}) =\tfrac{583}{179789679820800}$

with the numerators listed in {{OEIS2C|A272755}} and denominators in {{OEIS2C|A272757}}.

References

{{Citation | last1=Fabius | first1=J. | title=A probabilistic example of a nowhere analytic {{math|C{{hsp}}^∞}}-function | mr=0197656 | year=1966 | journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume=5 | issue=2 | pages=173–174 | doi=10.1007/bf00536652| s2cid=122126180 }}
{{Citation | last1=Jessen | first1=Børge | last2=Wintner|first2=Aurel| title=Distribution functions and the Riemann zeta function | mr=1501802 | year=1935 | journal=Trans. Amer. Math. Soc. | volume=38 | pages=48–88 | doi=10.1090/S0002-9947-1935-1501802-5 | doi-access=free }}
{{cite thesis|first1=Youri |last1=Dimitrov |title=Polynomially-divided solutions of bipartite self-differential functional equations

|year= 2006 |url= http://rave.ohiolink.edu/etdc/view?acc_num=osu1155149204}}

{{cite arXiv|first1=Juan|last1=Arias de Reyna|eprint=1702.06487|title=Arithmetic of the Fabius function|year=2017|class=math.NT }}
{{cite arXiv|first1=Juan|last1=Arias de Reyna|eprint=1702.05442|title=An infinitely differentiable function with compact support: Definition and properties|year=2017|class=math.CA}} (an English translation of the author's paper published in Spanish in 1982)

Category:Types of functions

Alkauskas, Giedrius (2001), "Dirichlet series associated with Thue-Morse sequence", [https://web.archive.org/web/20180412220529/https://www.pdf-archive.com/2018/04/13/thue-morse/thue-morse.pdf preprint].
Rvachev, V. L., Rvachev, V. A., "Non-classical methods of the approximation theory in boundary value problems", Naukova Dumka, Kiev (1979) (in Russian).