Fabius function

In mathematics, the Fabius function is a smooth cumulative distribution function F: \R \to [0,1] that is strictly increasing on [0,1] but nowhere real-analytic. It is the distribution function of the random series X = \sum_{n=1}^{\infty} 2^{-n}U_n, where the U_n are independent random variables uniformly distributed on [0,1] . Equivalently, it is characterized by F(0) = 0,\; F(1-x) = 1-F(x),\; F'(x) = 2F(2x)\,\left(0\leq x\leq\tfrac12\right). Although every positive-order derivative of F vanishes at both endpoints, F is nonconstant on every subinterval of [0,1] . The same object occurs, after a change of variables, as Rvachev's up function u , an even C^\infty bump function supported on [-1,1] . Its Fourier transform is an infinite product of sinc functions, or equivalently a product of powers of cosines. The function also has an unusual arithmetic side: every value F(a/2^n) at a dyadic rational is rational, and the numerators, denominators and 2-adic valuations of these values obey strong divisibility laws. The underlying distribution already appeared in work of Børge Jessen and Aurel Wintner in 1935. Jaap Fabius independently introduced it probabilistically in 1966 as an example of a nowhere-analytic smooth function, and V. A. Rvachev independently introduced the associated compactly supported solution of a functional–differential equation.…

In mathematics, the Fabius function is a smooth cumulative distribution function F: \R \to [0,1] that is strictly increasing on [0,1] but nowhere real-analytic. It is the distribution function of the random series X = \sum_{n=1}^{\infty} 2^{-n}U_n, where the U_n are independent random variables uniformly distributed on [0,1] . Equivalently, it is characterized by F(0) = 0,\; F(1-x) = 1-F(x),\; F'(x) = 2F(2x)\,\left(0\leq x\leq\tfrac12\right). Although every positive-order derivative of F vanishes at both endpoints, F is nonconstant on every subinterval of [0,1] . The same object occurs, after a change of variables, as Rvachev's up function u , an even C^\infty bump function supported on [-1,1] . Its Fourier transform is an infinite product of sinc functions, or equivalently a product of powers of cosines. The function also has an unusual arithmetic side: every value F(a/2^n) at a dyadic rational is rational, and the numerators, denominators and 2-adic valuations of these values obey strong divisibility laws. The underlying distribution already appeared in work of Børge Jessen and Aurel Wintner in 1935. Jaap Fabius independently introduced it probabilistically in 1966 as an example of a nowhere-analytic smooth function, and V. A. Rvachev independently introduced the associated compactly supported solution of a functional–differential equation. Near zero, F is smaller than every power of its argument; its exact rate of decay is governed by the lower branch of the Lambert W function together with a small correction that repeats each time the argument is halved. That correction alters F by only about two parts in a million, but it does not fade away as the argument tends to zero, so no asymptotic formula that omits it can be exact. The literature uses several incompatible conventions. In this article, F denotes the bounded cumulative distribution function, extended by constants outside [0,1] ; u denotes Rvachev's compactly supported up function; and \mathcal F denotes the signed global extension satisfying \mathcal F'(x)=2\mathcal F(2x) on the whole real line. Logarithms are natural unless a base is written explicitly, s_2(n) is the sum of the binary digits of n , and \nu_2(n) is the exponent of 2 in n . F(x)=0\quad(x\leq0),\qquad F(x)=1\quad(x\geq1), and, for 0\leq x\leq1 , F(1-x)=1-F(x), while F'(x) = 2F(2x) for 0\leq x\leq 1/2. The constant continuation outside the unit interval is smooth because all positive-order derivatives vanish at the endpoints. By symmetry, the differential equation can be written on the whole unit interval as F'(x)= \begin{cases} 2F(2x),&0\leq x\leq\tfrac12,\\[3pt] 2F(2-2x),&\tfrac12\leq x\leq1. \end{cases} The derivative is positive on (0,1) , so F is strictly increasing there. In particular, F\!\left(\tfrac12\right)=\tfrac12,\qquad F'\!\left(\tfrac12\right)=2, and F'(1-x)=F'(x),\qquad F'(x)+F'\!\left(\tfrac12-x\right)=2 \quad\left(0\leq x\leq\tfrac12\right). The endpoint flatness is F^{(m)}(0)=F^{(m)}(1)=0\qquad(m\geq1). Integrating the differential equation gives a self-integral equation, which together with the symmetry and the normalization characterizes F : F(x)=\int_0^{2x}F(t)\,dt \qquad\left(0\leq x\leq\tfrac12\right). X=\sum_{n=1}^{\infty}2^{-n}U_n converges almost surely and takes values in [0,1] . The Fabius function is F(x)=\Pr(X\leq x). The decomposition X\ \stackrel{d}{=}\ \frac{U+X'}2, where U is uniform on [0,1] and X' is an independent copy of X , gives F(x)=\int_0^1 F(2x-t)\,dt. For 0\leq x\leq\tfrac12 , the constant continuation F(t)=0 for t\leq0 reduces this to the self-integral equation above. The elementary moments of X are \operatorname E X=\frac12,\qquad \operatorname{Var}(X)=\frac1{36}. Set Y=2X-1=\sum_{n=1}^{\infty}2^{-n}V_n, \qquad V_n=2U_n-1. The variables V_n are independent and uniform on [-1,1] . The random variable Y has density u , and F'(x)=2u(2x-1)\qquad(0\leq x\leq1). Thus u is the infinite convolution u=h_1*h_2*h_3*\cdots,\qquad h_n(t)=2^{n-1}\mathbf 1_{[-2^{-n},\,2^{-n}]}(t), where \mathbf 1_A denotes the indicator of A . This construction accounts for both the compact support and the product formula for the Fourier transform. u(t)= \begin{cases} F(t+1),&-1\leq t\leq0,\\ F(1-t),&0\leq t\leq1,\\ 0,&|t|>1. \end{cases} Equivalently, u(t)=F(1-|t|)\qquad(|t|\leq1). It is even, nonnegative, positive on (-1,1) , satisfies u(0)=1 , has integral one, and has topological support exactly [-1,1] . Up to a constant factor it is the unique smooth compactly supported solution of the delay differential equation u'(t)=2\bigl(u(2t+1)-u(2t-1)\bigr), and the normalization \textstyle\int_{\R}u(t)\,dt=1 fixes that factor. Every derivative of u vanishes at t=\pm1 . \widehat u(z)=\int_{\mathbb R}u(t)e^{-2\pi izt}\,dt. Then \widehat u is an entire even function with \widehat u(0)=1 , and has the equivalent product representations \widehat u(z) = \prod_{h=0}^{\infty} \frac{\sin(\pi z/2^h)}{\pi z/2^h} =\prod_{m=1}^{\infty} \left(\cos\frac{\pi z}{2^m}\right)^m =\prod_{m=1}^{\infty} \left(1-\frac{z^2}{m^2}\right)^{1+\nu_2(m)}. Consequently, the zeros of \widehat u are precisely the non-zero integers, and the zero at a non-zero integer m has multiplicity 1+\nu_2(|m|) . Fourier inversion gives u(x)=\int_{\mathbb R}\widehat u(t)e^{2\pi itx}\,dt. Because u is smooth and compactly supported, its Fourier transform decreases faster than every inverse power on the real axis. \sum_{k\in\mathbb Z}u(x+k)=1. More generally, for every positive integer n , \sum_{k\in\mathbb Z}u\!\left(x+\frac{k}{n}\right)=n. Each sum is locally finite because u is compactly supported. These identities also follow from the zeros of \widehat u and the Poisson summation formula. The 2-periodic extension of u restricted to the interval [-1,1] has the absolutely and uniformly convergent cosine series u(x)=\frac12+ \sum_{j=0}^{\infty} \widehat u\!\left(\frac{2j+1}{2}\right) \cos\bigl((2j+1)\pi x\bigr). Only odd harmonics occur because \widehat u(k)=0 for every nonzero integer k . u(x)=\sum_{n=0}^{\infty}a_nP_{2n}(x), \qquad -1\leq x\leq1, where a_n=\frac{4n+1}{2}\int_{-1}^{1}u(x)P_{2n}(x)\,dx. A formula for the coefficients in terms of reciprocal-power values of F is a_n= 4^{-n}(4n+1) \sum_{k=0}^{n}(-1)^{n+k} \binom{2n}{n+k}\binom{2n+2k}{2n}(2k)!\, 2^{\binom{2k+1}{2}}F\!\left(2^{-2k-1}\right). The series converges absolutely and uniformly on [-1,1] , including at the endpoints. \mathcal F'(x)=2\mathcal F(2x)\qquad(x\in\mathbb R). It is given by \mathcal F(x)=0 for x\leq0 , by \mathcal F(x+1)=1-\mathcal F(x) for 0\leq x\leq1 , and by \mathcal F(x+2^r)=-\mathcal F(x)\qquad(0\leq x\leq2^r) for every positive integer r . Equivalently, it is the locally finite sum \mathcal F(x)= \sum_{n=0}^{\infty}(-1)^{s_2(n)} u(x-2n-1). It agrees with the bounded Fabius function on [0,1] , and on the interval [2n,2n+2] is a translated copy of u with sign (-1)^{s_2(n)} . Its sequence of signs is therefore the Thue–Morse sequence. Repeated differentiation gives \mathcal F^{(k)}(x) =2^{\binom{k+1}{2}}\mathcal F(2^kx) \qquad(k\geq0). In particular, every positive-order derivative vanishes at every integer. If q=a/2^n>0 is a positive dyadic number, with a a positive integer, then \mathcal F^{(k)}(q) =2^{\binom{k+1}{2}}\mathcal F(2^kq). For k>n , the argument 2^kq is an even integer, so these derivatives vanish. Hence the Taylor series of \mathcal F at a dyadic point is a polynomial of degree at most n ; when a is odd, the degree is exactly n . The function does not agree locally with this polynomial, and the dyadic rationals are dense, so F is nowhere real-analytic on [0,1] . c_n=\int_{-1}^{1}t^{2n}u(t)\,dt =\operatorname E(Y^{2n}) \qquad(n\geq0). All odd moments of u vanish. The even moments are rational and satisfy c_0=1,\qquad (2n+1)(2^{2n}-1)c_n =\sum_{k=0}^{n-1}\binom{2n+1}{2k}c_k \quad(n\geq1). The moment expansion of the Fourier transform is \widehat u(z)= \sum_{n=0}^{\infty} (-1)^n\frac{c_n}{(2n)!}(2\pi z)^{2n}. The moments are also related to Fabius values by c_n= 2(2n)!\,2^{\binom{2n+1}{2}} F\!\left(2^{-2n-1}\right). The moment-generating function of X is \begin{aligned} D(z) &=\operatorname E(e^{zX}) =\sum_{n=0}^{\infty}d_n\frac{z^n}{n!}\\ &=\prod_{j=1}^{\infty} \frac{e^{z/2^j}-1}{z/2^j}\\ &=e^{z/2}\widehat u\!\left(\frac{iz}{4\pi}\right)\\ &=1+z\int_0^1e^{zt}u(t)\,dt . \end{aligned} It obeys the dilation equation D(2z)=\frac{e^z-1}{z}D(z). Consequently, d_0 = 1 and (n+1)(2^n-1)d_n =\sum_{k=0}^{n-1}\binom{n+1}{k}d_k. for all n\geq 1 . The coefficients are the raw moments d_n=\operatorname E(X^n) and satisfy the identity d_n=n!\,2^{\binom n2}F(2^{-n}), which yields the values F(2^{-n}) in exact rational arithmetic. \begin{aligned} F\!\left(\frac{a}{2^n}\right) ={}&\frac{2^{-\binom{n+1}{2}}}{n!} \sum_{h=0}^{a-1}(-1)^{s_2(h)} \sum_{k=0}^{\lfloor n/2\rfloor} \binom n{2k} (2a-2h-1)^{n-2k}c_k . \end{aligned} The appearance of the binary digit sum s_2(h) reflects the Thue–Morse signs of the global extension. There is also an efficient recursive evaluator: given 0 , choose the unique integer n\geq0 such that 2^{-n}\leq x and put y=x-2^{-n} . Then F(x)=-F(y)+ \sum_{k=0}^{n} \frac{2^{\binom{k+1}{2}}F(2^{k-n})}{k!}\,y^k. For a dyadic input, replacing x by y removes its leading nonzero binary digit, so the iteration terminates after as many steps as there are 1s in the binary numerator, and every step is exact rational arithmetic. F(x)=o(x^N)\qquad(x\to0^+) for every fixed N>0 , and more precisely \log F(x)\sim-\frac{\log^2x}{2\log2}. A sharp description requires both the Lambert W function and a logarithmically periodic correction. Set L=\log2 and C_0=\frac{6\gamma^2+12\gamma_1-\pi^2}{12}, where \gamma is the Euler–Mascheroni constant and \gamma_1 is the first Stieltjes constant, normalized by \zeta(s)=\frac1{s-1}+\gamma-\gamma_1(s-1)+\cdots For sufficiently small positive x , set W=W_{-1}(-Lx),\qquad \lambda(x)=-\frac{W}{L}, where W_{-1} is the lower real branch of the Lambert W function; equivalently, \lambda(x) is the large positive solution of \lambda\,2^{-\lambda}=x . The sharp main term is M(x)= -\frac{7L}{12}-\frac12\log(\pi x) +\frac{C_0-W-\tfrac12W^2}{L} +\Psi\!\bigl(\lambda(x)\bigr), where \Psi is the one-periodic function described below, and, as x\to0^+ , \log F(x)=M(x)+O\!\left(\frac1{-\log x}\right). The function \Psi is real-valued, smooth and one-periodic, and is normalized to have mean zero over a period. With \chi_k=2\pi ik/L , its absolutely convergent Fourier series is \Psi(t)= \sum_{k\in\mathbb Z\setminus\{0\}} \left( -\frac{\Gamma(-\chi_k)\zeta(1-\chi_k)}{L} \right)e^{2\pi ikt}, where \Gamma is the gamma function and \zeta is the Riemann zeta function. The coefficient of index k is nonzero for every k\neq0 , because the gamma function has no zeros and the zeta function has no zeros on the line \operatorname{Re}(s)=1 , on which 1-\chi_k lies. Hence \Psi is nonconstant, and deleting \Psi(\lambda(x)) from M(x) leaves a bounded oscillation in \log F(x) that the O(1/(-\log x)) remainder cannot absorb. The oscillation is small: \Psi has amplitude about 2.1\times10^{-6} . The Lambert phase can itself be expanded in powers of -\log x and \log(-\log x) . Writing \ell=\log x , a=\log(-\ell) and b=\log L , the sharp form becomes \begin{aligned} \log F(x)= -\frac{\ell^2}{2L} +\frac{\ell a}{L} & -\left(\frac12+\frac{1+b}{L}\right)\ell -\frac{a^2}{2L} +\frac{ba}{L}\\[5pt] & +\left( \frac{6\gamma^2+12\gamma_1-\pi^2-6b^2}{12L} -\frac{7L}{12}-\frac{\log\pi}{2} \right) + \frac{a^2}{2L\ell} -\frac{ba}{L\ell} +\Psi\!\bigl(\lambda(x)\bigr) +O\!\left(\frac1{-\ell}\right). \end{aligned} At this precision, the periodic term is evaluated at the exact phase \lambda(x) . For reciprocal powers of two, let \lambda_n=\lambda(2^{-n}) , so that \lambda_n-\log_2\lambda_n=n , and set C_{\mathrm{sharp}}= \frac{6\gamma^2+12\gamma_1-\pi^2}{12L} -\frac{7L}{12}-\frac{\log\pi}{2}. Then \begin{aligned} \log F(2^{-n})={}& -\frac{L}{2}n^2-n\log n +\left(1+\frac L2\right)n -\frac{\log^2n}{2L} +C_{\mathrm{sharp}}\\ &-\frac{\log^2n}{2L^2n} +\Psi(\lambda_n) +O\!\left(\frac1n\right). \end{aligned} The oscillation is periodic in the exact Lambert phase, not in the integer n itself. D(-s)= \prod_{j=1}^{\infty} \frac{1-e^{-s/2^j}}{s/2^j}, whose logarithm has an exact dyadic-scale decomposition \log D(-s)= -\frac{\log^2s}{2L} +\frac12\log s +R(\log_2s)+E(s), with R one-periodic and |E(s)|\leq e^{-s}/(1-e^{-s})^2 . Rescaling by a factor of 2 therefore contributes an additive period-one term in the logarithmic variable, and saddle-point inversion carries the centred form of that term to \Psi(\lambda(x)) in the asymptotic of F(x) . \log F(x)= M(x)+ \sum_{j=1}^{N-1} \frac{P_j(\lambda(x))}{\lambda(x)^j} +O\!\left(\lambda(x)^{-N}\right), where M is the sharp main term. The first correction is P_1(t)= \frac1{24}+\frac1{2L} -\frac{\Psi(t)+(\Psi'(t))^2}{2L^2}.

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