Kolmogorov–Arnold representation theorem

In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariable continuous function can be represented as a superposition of continuous functions of two variables. It solved a version of Hilbert's thirteenth problem.

The works of Kolmogorov and Arnold established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition.

More specifically

$f(\vec x) = \sum_{q=0}^{2n} \Phi_{q}\left(\sum_{p=1}^{n} \phi_{q,p}(x_{p})\right)$

Constructive proofs, and even more specific constructions can be found in

For example:

f(x,y) = xy can be written as f(x,y) = exp(log x + log y)

f(x,y,z) = x^y / z can be written as f(x, y, z) = exp(exp(log y + log log x) + (−log z))

In a sense, they showed that the only true multivariate function is the sum, since every other function can be written using univariate functions and summing.

Original references

• A. N. Kolmogorov, "On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables", Proceedings of the USSR Academy of Sciences, 108 (1956), pp. 179–182; English translation: Amer. Math. Soc. Transl., 17 (1961), pp. 369–373.
• V. I. Arnold, "On functions of three variables", Proceedings of the USSR Academy of Sciences, 114 (1957), pp. 679–681; English translation: Amer. Math. Soc. Transl., 28 (1963), pp. 51–54.

Further reading

• S. Ya. Khavinson, Best Approximation by Linear Superpositions (Approximate Nomography), AMS Translations of Mathematical Monographs (1997)