View source for Fabius function

{{short description|Nowhere analytic, infinitely differentiable function}}
[[File:Mplwp Fabius function.svg|thumb|upright=1.35|Graph of the Fabius function on the interval <nowiki>[0,1]</nowiki>.]]
[[File:Graph of the Fabius function.png|thumb|Extension of the function to the nonnegative real numbers.]]

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

This function satisfies the initial condition <math>f(0) = 0</math>, the symmetry condition <math>f(1-x) = 1 - f(x)</math>  for <math>0 \le x \le 1,</math> and the [[functional differential equation]] 

:<math>f'(x) = 2 f(2 x)</math> 

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

It was also written down as the [[Fourier transform]] of
 
:<math> \hat{f}(z) = \prod_{m=1}^\infty \left(\cos\frac{\pi z}{2^m}\right)^m</math>

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

:<math>\sum_{n=1}^\infty2^{-n}\xi_n,</math>

where the {{math|''&xi;''<sub>''n''</sub>}} are [[independence (probability)|independent]] [[uniform distribution (continuous)|uniformly distributed]] [[random variable]]s on the [[unit interval]]. That distribution has an expectation of <math>\tfrac{1}{2}</math> and a variance of <math>\tfrac{1}{36}</math>.

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<sup>''r''</sup>) = −''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 2<sup>''r''</sup>}} 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 ''Rvachev up function'' is closely related: {{math|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:<ref>{{Cite OEIS |A272755 |Numerators of the Fabius function F(1/2^n). }}</ref><ref>{{Cite OEIS |A272757 |Denominators of the Fabius function F(1/2^n). }}</ref>  

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

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

==References==
{{Reflist}}
*{{Citation | last1=Fabius | first1=J. | title=A probabilistic example of a nowhere analytic {{math|''C''{{hsp}}<sup>∞</sup>}}-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).


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