A Version of Hilbert's 13th Problem for Infinitely Differentiable Functions
© S. Akashi and S. Kodama. 2010
Received: 29 December 2009
Accepted: 4 March 2010
Published: 14 March 2010
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.
1. Superposition Representability and Superposition Irrepresentability
In 1957, Kolmogorov and Arnold  solved Hilbert's 13th problem asking if all continuous real-valued functions of several real variables can be represented as superpositions of continuous functions of fewer variables. Moreover, in 1964, Vitushkin  solved the problem, which had been derived from Hilbert's 13th problem, asking if all finitely differentiable real-valued functions of several real variables can be represented as superpositions of finitely differentiable functions of fewer variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.
Let (resp., ) be a set of functions of three variables (resp., two variables) satisfying the condition such as continuity or differentiability. Then, the superposition representability can be classified into the following two concepts.
Here, for a certain condition , is said to be strongly (resp., weakly) representable, if strong (resp., weak) representability under the condition holds. It is clear that is weakly representable, if is strongly representable. By the same way as above, the superposition irrepresentability can be also classified into the following two concepts.
Here, for a certain condition , is said to be strongly (resp., weakly) irrepresentable, if strong (resp., weak) irrepresentability under the condition holds. It is clear that is weakly irrepresentable, if is strongly irrepresentable. Moreover, it is also clear that is weakly irrepresentable (resp., representable), if is not strongly representable (resp., irrepresentable). Therefore, we can classify a condition such as continuity or differentiability into three cases.
For example, if we take continuity as an example of , then owing to Kolmogorov and Arnold, we can say that continuity satisfies Case 1. As for the proof, we refer to [1, 3]. Moreover, if we take analyticity as an example of , then, owing to Babenko, Erohin, and Akashi, we can say that analyticity satisfies Case 3. As for the proof, we refer to [4–7]. If we take finite differentiability as an example of , then, owing to Vituskin, we can say that finite differentiability satisfies Case 3. As for the proof, we refer to . It is clear that polynomial condition satisfies Case 2. In the following section, this result will be formulated as a generalized relation between polynomial condition and infinite differentiability condition.
Recently, it is discussed that Hilbert's 13th problem can be applied to the theory of multidimensional numerical data compression. Since the results stated above show that it is important for any functions of three variables to find the appropriate superpositions which can approximate most efficiently to the original function. Therefore, nonlinear theoretic approximation methods will play important roles in the theory of multidimensional numerical data compression. As for the nonlinear theoretic approximation methods, we can refer to Takahashi's results .
2. Weak Irrepresentability of Polynomial Condition
In this section, we prove that polynomial condition is weakly irrepresentable.
So, we have a contradiction.
This lemma shows that polynomial condition satisfies Case 2, because this condition is also weakly representable.
3. Strong Irrepresentability of Infinite Differentiability Condition
In this section, we prove that infinite differentiability condition is strongly irrepresentable.
There exists an infinitely differentiable function defined on with values in , which cannot be represented as any superposition constructed from several infinitely differentiable functions defined on with values in .
Actually, this equality shows that, for any positive integer , can be represented as a certain -time nested superposition constructed from several infinitely differentiable functions defined on . Therefore, we have a contradiction.
The authors would like to express their hearty thanks to the referee who has given several pieces of suggestive advice to make the original manuscript more comprehensible, and they are also so thankful to Professor Anthony To-Ming Lau for having communicated several times with us in the course of reviewing procedure.
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