A superposition theorem for bounded continuous functions
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- by Stephen Demko PDF
- Proc. Amer. Math. Soc. 66 (1977), 75-78 Request permission
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
- Raouf Doss, A superposition theorem for unbounded continuous functions, Trans. Amer. Math. Soc. 233 (1977), 197–203. MR 582781, DOI 10.1090/S0002-9947-1977-0582781-1 T. Hedberg, The Kolmogorov superposition theorem, Appendix II, Topics in Approximation Theory by H. S. Shapiro, Springer-Verlag, Berlin and New York, 1971.
- A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Amer. Math. Soc. Transl. (2) 28 (1963), 55–59. MR 0153799, DOI 10.1090/trans2/028/04
- G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 66 (1977), 75-78
- MSC: Primary 26A72
- DOI: https://doi.org/10.1090/S0002-9939-1977-0457651-5
- MathSciNet review: 0457651