The Kolmogorov–Arnold representation theorem is closely related to Hilbert's 13th problem. In his Paris lecture at the International Congress of Mathematicians in 1900, David Hilbert formulated 23 problems which in his opinion were important for the further development of mathematics. The 13th of these problems dealt with the solution of general equations of higher degrees. It is known that for algebraic equations of degree 4 the solution can be computed by formulae that only contain radicals and arithmetic operations. For higher orders, Galois theory shows us that the solutions of algebraic equations cannot be expressed in terms of basic algebraic operations. It follows from the so called Tschirnhaus transformation that the general algebraic equation can be translated to the form. The Tschirnhaus transformation is given by a formula containing only radicals and arithmetic operations and transforms. Therefore, the solution of an algebraic equation of degree can be represented as a superposition of functions of two variables if and as a superposition of functions of variables if. For the solution is a superposition of arithmetic operations, radicals, and the solution of the equation. A further simplification with algebraic transformations seems to be impossible which led to Hilbert's conjecture that "A solution of the general equation of degree 7 cannot be represented as a superposition of continuous functions of two variables". This explains the relation of Hilbert's thirteenth problem to the representation of a higher-dimensional function as superposition of lower-dimensional functions. In this context, it has stimulated many studies in the theory of functions and other related problems by different authors.
Variants
A variant of Kolmogorov's theorem that reduces the number of outer functions is due to George Lorentz. He showed in 1962 that the outer functions can be replaced by a single function. More precisely, Lorentz proved the existence of functions,, such that David Sprecher replaced the inner functions by one single inner function with an appropriate shift in its argument. He proved that there exist real values, a continuous function, and a real increasing continuous function with, for, such that Phillip A. Ostrand generalized the Kolmogorov superposition theorem to compact metric spaces. For let be compact metric spaces of finite dimension and let. Then there exists continuous functions and continuous functions such that any continuous function is representable in the form
