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A superposition theorem for bounded continuous functions

Author: Stephen Demko
Journal: Proc. Amer. Math. Soc. 66 (1977), 75-78
MSC: Primary 26A72
MathSciNet review: 0457651
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Abstract: It is shown that there exist $ 2n + 1$ real valued, continuous functions $ {\phi _0}, \ldots ,{\phi _{2n}}$ defined on $ {{\mathbf{R}}^n}$ such that every bounded real valued continuous function on $ {{\mathbf{R}}^n}$ is expressible in the form $ \Sigma _{i = 0}^{2n}g \circ {\phi _i}$ for some $ g \in C({\mathbf{R}})$. Extensions to some unbounded functions are also made.

References [Enhancements On Off] (What's this?)

  • [1] R. Doss, A superposition theorem for unbounded continuous functions, preprint. MR 0582781 (58:28366)
  • [2] T. Hedberg, The Kolmogorov superposition theorem, Appendix II, Topics in Approximation Theory by H. S. Shapiro, Springer-Verlag, Berlin and New York, 1971.
  • [3] A. N. Kolmogorov, On the representation of continuous functions of many variables by superpositions of continuous functions of one variable and addition, Amer. Math. Soc. Transl. 28 (1963), 55-59. MR 0153799 (27:3760)
  • [4] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 0213785 (35:4642)

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Keywords: Kolmogorov superposition theorem, Baire category theorem
Article copyright: © Copyright 1977 American Mathematical Society

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