Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 287647, 5 pages
Research Article

A Version of Hilbert's 13th Problem for Infinitely Differentiable Functions

Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda City, Chiba Prefecture 278-8510, Japan

Received 29 December 2009; Accepted 4 March 2010

Academic Editor: Anthony To Ming Lau

Copyright © 2010 Shigeo Akashi and Satoshi Kodama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.