Fabius function: Difference between revisions

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{{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}}. It was also written down as the Fourier transform of

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>

:<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>

Line 12 ⟶ 21:

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

where the {{math|''&xi;''_''n''}} 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^''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]].

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>

There is a unique extension of {{mvar|f}} to the real numbers that satisfies the same equation. 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''<ref>{{cite web | url=https://oeis.org/A288163 | title=A288163 - Oeis }}</ref> is closely related: <math display="block"> u(t)=\begin{cases} F(t+1),\quad |t|<1 \\ 0, \quad |t|\geq 1 \end{cases}</math> which fulfills the [[Delay differential equation]]<ref>{{cite arXiv | eprint=1702.06487 | author1=Juan Arias de Reyna | title=Arithmetic of the Fabius function | year=2017 | class=math.NT }}</ref>

<math display="block">\frac{d}{dt}u(t)=2u(2t+1)-2u(2t-1).</math> (see [[Delay differential equation|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:<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}}.

==Asymptotic==

<math>\begin{align}\log f(x)&=-\frac{\log^2x}{2\log2}+\frac{\log x\cdot\log(-\log x)}{\log2}-\left(\frac12+\frac{1+\log\log2}{\log2}\right)\log x -\frac{\log^2(-\log x)}{2\log2}+\frac{\log\log 2\cdot\log(-\log x)}{\log2}\\&+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2-6\log^2\log2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)+\frac{\log^2(-\log x)}{2\log2\cdot\log x}-\frac{\log\log2\cdot\log(-\log x)}{\log2\cdot\log x}+O\!\left(\frac1{\log x}\right)\end{align}</math>

for <math>x\to0^+,</math> where <math>\gamma</math> is [[Euler's constant]], and <math>\gamma_1</math> is the [[Stieltjes constants|Stieltjes constant]]. Equivalently,

<math>\log f\!\left(2^{-n}\right)=-\frac{n^2\log2}2-n\log n+\left(1+\frac{\log2}2\right)n -\frac{\log^2n}{2\log2}+\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^2n}{2n\log^22}+O\!\left(\frac1n\right)</math>

for <math>n\to\infty.</math>

==References==

{{Reflist}}

*{{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 arxivarXiv|first1=Jan Kristian Juan|last1=HauglandArias de Reyna|eprint=16091702.0799906487|title =EvaluatingArithmetic of the Fabius function|year=20162017|class=math.GMNT }}

*{{cite arxivarXiv|first1=Juan|last1=Arias de Reyna|eprint=1702.0648705442|title=ArithmeticAn ofinfinitely the Fabiusdifferentiable function with compact support: Definition and properties|year=2017|class=math.NT CA}} (an English translation of the author's paper published in Spanish in 1982)

*{{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].