 Kolmogorov–Arnold representation theorem

Every real continuous function on a compact square can be represented as ${\sum }_{q=1}^5{G}_q(a_q(x)+b_q(y))$, with continuous one-variable functions and fixed inner functions $a_q,b_q$ independent of the represented function. Grid-separated additive coordinates give the inner functions. For a residual $r$ of norm $M$, assign $r$ at a sample point divided by three to each separated rectangle image, and continuously interpolate between images. Each point lies in at least three rectangles, so if $k\in \{3,4,5\}$ summands are accurate, the error is at most $[|1-k/3|+(5-k)/3]M+5\eta /3=2M/3+5\eta /3$. Choose oscillation $\eta \le M/10$ to obtain contraction $5/6$. Iteration gives outer functions by uniform convergence of corrections bounded by $M(5/6{)}^j/3$. Whole-plane reduction for continuous superposition removes compactness of the original domain without assuming boundedness of the original function.

The Kolmogorov–Arnold representation theorem, also known as the Kolmogorov–Arnold function representation theorem, is a result in the theory of multivariate functions that provides a way to express any continuous multivariate function as a superposition of continuous functions of fewer variables.