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A superposition theorem for unbounded continuous functions


Author: Raouf Doss
Journal: Trans. Amer. Math. Soc. 233 (1977), 197-203
MSC: Primary 26A72
DOI: https://doi.org/10.1090/S0002-9947-1977-0582781-1
MathSciNet review: 0582781
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Abstract: Let $ {R^n}$ be the n-dimensional Euclidean space. We prove that there are 4n real functions $ {\varphi _q}$ continuous on $ {R^n}$ with the following property: Every real function f, not necessarily bounded, continuous on $ {R^n}$, can be written $ f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}$, where g, h are 2 real continuous functions of one variable, depending on f.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0582781-1
Keywords: Superposition of functions, several variables, Hilbert Problem 13
Article copyright: © Copyright 1977 American Mathematical Society

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