Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Constructive decomposition of a function of two variables as a sum of functions of one variable

Author: Eva Miliczká
Journal: Proc. Amer. Math. Soc. 137 (2009), 607-614
MSC (2000): Primary 26B40, 54C30; Secondary 54F99, 54C25
Published electronically: August 27, 2008
MathSciNet review: 2448582
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Abstract: Given a compact set $ K$ in the plane, which does not contain any triple of points forming a vertical and a horizontal segment, and a map $ f\in C(K)$, we give a construction of functions $ g,h\in C(\mathbb{R})$ such that $ f(x,y)=g(x)+h(y)$ for all $ (x,y)\in K$. This provides a constructive proof for a part of Sternfeld's theorem on basic embeddings in the plane. In our proof the set $ K$ is approximated by a finite set of points.

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Additional Information

Eva Miliczká
Affiliation: Institute of Computer Science, Faculty of Science, P. J. Šafárik University, Jesenná 5, 04001 Košice, Slovakia

Keywords: Basic embedding, plane compactum, Kolmogorov representation theorem, Hilbert's 13th problem, finite approximation of sets
Received by editor(s): January 16, 2007
Received by editor(s) in revised form: January 31, 2008
Published electronically: August 27, 2008
Additional Notes: The author was supported by grants VEGA 1/3002/06 and VEGA 1/3128/06
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.