Kolmogorov–Arnold representation theorem

In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function ${\displaystyle f\colon [0,1]^{n}\to \mathbb {R} }$ can be represented as a superposition of the two-argument addition of continuous functions of one variable. It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary.

The works of Vladimir Arnold and Andrey Kolmogorov 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,

${\displaystyle f(\mathbf {x} )=f(x_{1},\ldots ,x_{n})=\sum _{q=0}^{2n}\Phi _{q}\!\left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right)}$.

where ${\displaystyle \phi _{q,p}\colon [0,1]\to \mathbb {R} }$ and ${\displaystyle \Phi _{q}\colon \mathbb {R} \to \mathbb {R} }$.

There are proofs with specific constructions.

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.^: 180