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Linear superposition of smooth functions


Author: Robert Kaufman
Journal: Proc. Amer. Math. Soc. 46 (1974), 360-362
MSC: Primary 26A72; Secondary 46E15
DOI: https://doi.org/10.1090/S0002-9939-1974-0352374-2
MathSciNet review: 0352374
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Abstract: We give a simple proof of the impossibility of representing an arbitrary continuous function as a superposition (1), when $ {F_1}, \cdots ,{F_N}$ are smooth mappings of $ {R^{n + 1}}$ to $ {R^n}$. The main tool is the Riemann-Lebesgue lemma.


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  • [1] B. L. Fridman, An improvement in the smoothness of functions in A. N. Kolmogorov's theorem on superposition, Dokl. Akad. Nauk SSSR 177 (1967), 1019 -1022 = Soviet Math. Dokl. 8 (1967), 1550-1553. MR 37 #663. MR 0225066 (37:663)
  • [2] -, A nowhere dense space of linear superpositions of functions of several variables, Izv. Akad. Nauk SSSR Ser Mat. 36 (1972), 814-846 = Math. USSR Izv. 6 (1972), 807-837. MR 0318422 (47:6969)
  • [3] T. Hedberg, The Kolmogorov superposition theorem, Topics in Approximation Theory, Appendix II, Lecture Notes in Math., vol. 187, Springer-Verlag, Berlin and New York, 1971, pp. 267-275. MR 0437981 (55:10902)
  • [4] D. A. Sprecher, An improvement in the superposition theorem of Kolmogorov, J. Math. Anal. Appl. 38 (1972), 208-213. MR 46 #1981. MR 0302838 (46:1981)
  • [5] A. G. Vituškin and G. M. Henkin, Linear superpositions of functions, Uspehi Mat. Nauk 22 (1967), no. 1 (133), 77-124 = Russian Math. Surveys 22 (1967), no. 1, 77-125. MR 38 #6010. MR 0237729 (38:6010)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0352374-2
Keywords: Smooth functions, Kolmogorov superposition theorem, Baire category
Article copyright: © Copyright 1974 American Mathematical Society

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