Nomographic functions are nowhere dense

Author:
R. Creighton Buck

Journal:
Proc. Amer. Math. Soc. **85** (1982), 195-199

MSC:
Primary 41A63; Secondary 41A30

DOI:
https://doi.org/10.1090/S0002-9939-1982-0652441-4

MathSciNet review:
652441

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Abstract | References | Similar Articles | Additional Information

Abstract: A function of variables is nomographic if it can be represented in the format

**[1]**V. I. Arnol′d,*On the representability of a function of two variables in the form 𝜒[𝜙(𝑥)+𝜓(𝑦)]*, Uspehi Mat. Nauk (N.S.)**12**(1957), no. 2(74), 119–121 (Russian). MR**0090623****[2]**V. I. Arnol′d,*Some questions on approximation and representation of functions*, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 339–348 (Russian). MR**0121454****[3]**R. C. Buck,*Approximate complexity and functional representation*, J. Math. Anal. Appl.**70**(1979), no. 1, 280–298. MR**541075**, https://doi.org/10.1016/0022-247X(79)90091-X**[4]**R. Creighton Buck,*Characterization of classes of functions*, Amer. Math. Monthly**88**(1981), no. 2, 139–142. MR**606252**, https://doi.org/10.2307/2321136**[5]**Raouf Doss,*On the representation of the continuous functions of two variables by means of addition and continuous functions of one variable*, Colloq. Math.**10**(1963), 249–259. MR**0155949**, https://doi.org/10.4064/cm-10-2-249-259**[6]**B. L. Fridman,*Nowhere denseness of the space of linear superpositions of functions of several variables*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 814–846 (Russian). MR**0318422****[7]**G. M. Henkin,*Linear superpositions of continuously differentiable functions*, Dokl. Akad. Nauk SSSR**157**(1964), 288–290 (Russian). MR**0166319****[8]**Robert Kaufman,*Linear superposition of smooth functions*, Proc. Amer. Math. Soc.**46**(1974), 360–362. MR**0352374**, https://doi.org/10.1090/S0002-9939-1974-0352374-2**[9]**A. N. Kolmogorov,*On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition*, Dokl. Akad. Nauk SSSR**114**(1957), 953–956 (Russian). MR**0111809****[10]**G. G. Lorentz,*The 13th problem of Hilbert*, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 419–430. MR**0507425****[11]**David A. Sprecher,*A survey of solved and unsolved problems on superpositions of functions*, J. Approximation Theory**6**(1972), 123–134. Collection of articles dedicated to J. L. Walsh on his 75th birthday, VI (Proc. Internat. Conf. Approximation Theory, Related Topics and their Applications, Univ. Maryland, College Park, Md., 1970). MR**0348347**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0652441-4

Keywords:
Superpositions,
Hilbert,
nowhere dense

Article copyright:
© Copyright 1982
American Mathematical Society