Linear superpositions with mappings which lower dimension

Author:
Y. Sternfeld

Journal:
Trans. Amer. Math. Soc. **277** (1983), 529-543

MSC:
Primary 26B40; Secondary 54F45

DOI:
https://doi.org/10.1090/S0002-9947-1983-0694374-9

MathSciNet review:
694374

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for every -dimensional compact metric space , there exist functions in and mappings on with -dimensional range each, with the following property: for every , every tuple of the 's and every tuple of the 's, each can be represented as , with and .

It is also shown that in many cases the number is the smallest possible.

**[1]**V. I. Arnold,*Representation of continuous functions of three variables*, Amer. Math. Soc. Transl. (2)**28**(1963), 61-149. MR**0153797 (27:3758)****[2]**C. F. K. Jung,*Mappings on compact metric spaces*, Colloq. Math.**19**(1968), 73-76. MR**0229215 (37:4789)****[3]**J. Keesling,*Closed mappings which lower dimension*, Colloq. Math.**20**(1969), 237-241. MR**0248768 (40:2019)****[4]**A. N. Kolmogorov,*On the representation of continuous functions of many variables by superpositions of continuous functions of one variable and addition*, Dokl. Akad. Nauk SSSR**114**(1959), 953-956. MR**0111809 (22:2669)****[5]**K. Kuratowski,*Topology*. II, Academic Press, New York, 1968.**[6]**A. Lelek,*On mappings that change dimension of spheres*, Colloq. Math.**10**(1963), 45-48. MR**0152990 (27:2961)****[7]**K. Menger,*Kurventheorie*, Teubner, Leipzig, 1932.**[8]**P. A. Ostrand,*Dimension of metric spaces and Hubert's problem*, Bull. Amer. Math. Soc.**7**(1965), 619-622. MR**0177391 (31:1654)****[9]**Y. Sternfeld,*Uniformly separating families of functions*, Israel. Math.**29**(1978), 61-91. MR**0487991 (58:7570)****[10]**-,*Superpositions of continuous functions*, J. Approx. Theory**25**(1979), 360-368. MR**535937 (82d:26008)****[11]**-,*Dimension of subsets of product spaces*, Proc. Amer. Math. Soc.**82**(1981), 452-454. MR**612738 (83a:54054)****[12]**-,*Uniformly separating families of functions*. II, preprint.

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0694374-9

Article copyright:
© Copyright 1983
American Mathematical Society