Theorem 1

For any integer \(n\geq 2\) there are continuous real functions \(\psi^{p,q}(x)\) on the closed unit interval \(E^{1} = [0,1]\) such that each continuous real function \(f(x_{1},\cdots,x_{n})\) on the \(n\)-dimensional unit cube \(E^{n}\) is representable as

\[f(x_{1},\cdots,x_{n}) = \sum_{q=1}^{2n+1} \chi^{q} \left[\sum_{p=1}^{n} \psi^{p,q}(x_{p})\right]\]

where \(\chi^{q}(y)\) are continuous real function.


Step 1: Construction of the functions \(\psi^{pq}\).

Consider the closed intervals

\[\begin{aligned} & A_{k,i}^{q} = \left[\frac{1}{(9n)^{k}}\left(i - 1 - \frac{q}{3n}\right), \frac{1}{(9n)^{k}}\left(i - \frac{1}{3n} - \frac{q}{3n}\right)\right],\\ &\qquad \qquad \qquad \qquad 1\leq i \leq (9n)^{k} +1, 1\leq q \leq 2n +1, k= 1,2,\cdots \end{aligned}\]

with lengths \(\frac{1}{9n}^{k} ( 1 - \frac{1}{3n})\) and Accordingly, for fixed \(k\) and \(q\) by passing \(i\) to \(i+1\) using a shift to the right over a distance \(1/(9n)^{k}\). Accordingly, for fixed \(k\) and \(q\) the cubes

\[S_{k,i_{1},\cdots,i_{n}}^{q} = \prod_{p=1}^{n} A_{k,i_{p}}^{q}\]

with edges of lengths \(\frac{1}{(9n)^{k}}\) cover the unit cube \(E^{n}\) to within the separating slits of widths \(\frac{1}{3n(9n)^{k}}\). It's easy to verify the following

Lemma 1

The system of all cubes \(S_{k,i_{1},\cdots,i_{n}}^{q}\) with constant \(k\) and variable \(q\) and \(i_{1},\cdots,i_{n}\) covers the unit cube \(E^{n}\) so that each point belonging to \(E^{n}\) is covered by at least \(n+1\) times. \(\label{lem 1}\)


Using induction on \(k\) we can prove the following

Lemma 2

There exist constants \(\lambda^{pq}_{k,i}\) and \(\epsilon_{k}\) such that

  1. \(\lambda^{pq}_{k,i} \leq \lambda^{pq}_{k,i+1} \leq \lambda^{pq}_{k,i} + 1/2^{k}\);

  2. \(\lambda^{pq}_{k,i} \leq \lambda^{pq}_{k+1,i^{\prime}} \leq \lambda^{pq}_{k,i} + \epsilon_{k+1} - \epsilon_{k}\) if the closed intevals \(A_{k,i}^{q}\) and \(A_{k+1,i^{\prime}}^{q}\) do not intersect;

  3. the closed interval \(\Delta^{q}_{k,i_{1}, \cdots, i_{n}} = \left[\sum_{p}\lambda^{pq}_{k,i_{p}}, \sum_{p}\lambda^{pq}_{k,i_{p}} + n \epsilon_{k}\right]\) are pairwise disjoint for fixed \(k\) and \(q\).

It's easy to note that 1. and 3. imply

  1. \(\epsilon \leq 1/2^{k}\).

Lemma 3

For fixed \(p\) and \(q\) the condition

  1. \(\lambda^{pq}_{k,i} \leq \psi^{pq}(x) \leq \lambda^{pq}_{k,i} + \epsilon_{k}\) uniquely determine a continuous function \(\psi^{pq}\) on \(E^{1}\).

From 5. and 3. it follows that

  1. \(\sum_{p} \psi^{pq} (x_{p}) \in \Delta^{q}_{k,i_{1}, \cdots, i_{n}}\) for \((x_{1}, \cdots, x_{n}) \in S_{k,i_{1},\cdots,i_{n}}^{q}\).


Step 2: Construction of the functions \(\chi^{q}\).

Let \(\chi_{0}^{q} \equiv 0\), while for \(r >0\), \(\chi_{r}^{q}\) will be defined by induction on \(r\) simultaneously with the natural number \(k_{r}\).

Denote as below

\[\begin{aligned}f_{r}(x_{1},\cdots,x_{n}) = & \sum_{q=1}^{2n+1} \chi_{r}^{q}\left[\sum_{p} \psi^{pq}(x_{p})\right]\\ M_{r}= & \sup_{E^{n}} \left|f -f_{r}\right|\end{aligned}\]

Inductive step: Assuming \(\chi_{r-1}^{q}\) and \(k_{r-1}\) have already been determined.

Passing to step \(r\):

Since the diameter of the cubes \(S_{k,i_{1},\cdots,i_{n}}^{q}\) tend to zero as \(k \to \infty\), we can choose \(k_{r}\) so large that the oscillation of the difference \(f - f_{r-1}\) does not exceed \(M_{r}/(2n+2)\) on any \(S_{k_{r},i_{1},\cdots,i_{n}}^{q}\).

Let \(\xi_{k,i}^{q}\) be arbitrary points belonging to the corresponding closed intervals \(A_{k,i}^{q}\). For the closed interval \(\Delta^{q}_{k,i_{1}, \cdots, i_{n}}\) we put

\[\begin{aligned}\chi_{r}^{q}(y) = & \chi_{r-1}^{q}(y) + \frac{1}{n+1} \left[f(\xi_{k,i_{1}}^{q}, \cdots, \xi_{k,i_{n}}^{q}) - f_{r}(\xi_{k,i_{1}}^{q}, \cdots, \xi_{k,i_{n}}^{q})\right]\\ & \left|\chi_{r}^{q} (y) - \chi_{r-1}(y)\right| \leq \frac{1}{n+1} M_{r-1}\end{aligned}\]

Outside the closed intervals \(\Delta^{q}_{k,i_{1}, \cdots, i_{n}}\) the function \(\chi_{r}^{q}\) is defined arbitrarily, with the preservation of the inequality above and continuity.

\[\begin{aligned}f(x_{1},\cdots,x_{n}) - f_{r}(x_{1},\cdots,x_{n}) = f(x_{1},\cdots,x_{n}) - f_{r-1}(x_{1},\cdots,x_{n})\\ - \sum_{q=1}^{2n+1} \left\{\chi_{r}^{q}\left[\sum_{p}\psi^{pq}(x_{p})\right] - \chi_{r-1}^{q}\left[\sum_{p}\psi^{pq}(x_{p})\right]\right\}\end{aligned}\]

We represent the sum in above equation in the form \(\sum^{\prime} + \sum^{\prime\prime}\), where the sum \(\sum^{\prime}\) extends over certain \(n+1\) values of \(q\) for which the point \((x_{1},\cdots,x_{n})\) is contained in one of the cubes \(S_{k_{r},i_{1},\cdots,i_{n}}^{q}\) (by Lemma 1\(\ref{lem 1}\), such cubes exist) and the sum \(\sum^{\prime\prime}\) extends over the remaining \(n\) values of \(q\). Hence, for each term in \(\sum^{\prime}\) we have

\[\begin{aligned} &\chi_{r}^{q}\left[\sum_{p}\psi^{pq}(x_{p})\right] - \chi_{r-1}^{q}\left[\sum_{p}\psi^{pq}(x_{p})\right] \\ = \frac{1}{n+1} &\left[f(\xi_{k,i_{1}}^{q}, \cdots, \xi_{k,i_{n}}^{q}) - f_{r-1}(\xi_{k,i_{1}}^{q}, \cdots, \xi_{k,i_{n}}^{q})\right] \\ = \frac{1}{n+1} & \left[f(x_{1},\cdots,x_{n}) - f_{r-1}(x_{1},\cdots,x_{n})\right] + \frac{\omega^{q}}{n+1} \end{aligned}\]

where

\[|\omega^{q}| \leq \frac{1}{2n+2}M_{r}\]

which implies

\[\begin{aligned}|f - f_{r}| = \left|\frac{1}{n+1} \sum^{\prime}\omega^{q} + \sum^{\prime \prime} (\chi_{r}^{q} - \chi_{r-1}^{q})\right| \leq \\ \frac{1}{2n+2} M_{r-1} + \frac{n}{n+1} M_{r-1} = \frac{2n +1}{2n +2} M_{r-1} \end{aligned}\]

Since inequality holds at any point \((x_{1},\cdots,x_{n}) \in E^{n}\), it follows that the absolute values of the differences \(\chi^{q} - \chi_{r}^{q}\) do not exceed the corresponding terms of the absolutely convergent series

\[\sum_{r} \frac{1}{n+1} M_{r-1}\]

Therefore the functions \(\chi_{r}^{q}\) converges uniformly to continuous limit functions \(\chi^{q}\) for \(r\to \infty\).


Reference:

[1] Kolmogorov, A. N. (1957). On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition. (in Russian) In Doklady Akademii Nauk (Vol. 114, No. 5, pp. 953-956). Russian Academy of Sciences.

[2] Arnold, V. I. (1957). On functions of three variables. (in Russian) Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965, 5-8.