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 continuous single-variable functions.

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 ${\displaystyle \Phi _{q}\colon \mathbb {R} \to \mathbb {R} }$.

In this representation, the inner functions ${\displaystyle \phi _{q,p}}$ are continuous and universal, that is, independent of ${\displaystyle f}$, while the outer functions ${\displaystyle \Phi _{q}}$ depend on the specific function being represented. The same representation formula extends to all multivariate functions ${\displaystyle f}$, including discontinuous ones, as discussed in . If ${\displaystyle f}$ is continuous, then the corresponding outer functions ${\displaystyle \Phi _{q}}$ are continuous; if ${\displaystyle f}$ is discontinuous, the outer functions are generally discontinuous, while the inner functions ${\displaystyle \phi _{q,p}}$ remain unchanged, being the same universal functions.

There are proofs with specific constructions.

It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary. In a sense, they showed that the only true continuous multivariate function is the sum, since every other continuous function can be written using univariate continuous functions and summing.