Lesson plan
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.…
Article summary
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.…
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