MATHEMATICS

V. I. ARNOLD

ON FUNCTIONS OF THREE VARIABLES

(Presented by Academician A. N. Kolmogorov, 10 IV 1957)

Below we briefly indicate a method of proof of a theorem which gives a complete solution of Hilbert’s 13th problem (in the sense of refuting the hypothesis stated by Hilbert).

Theorem 1. Every prescribed real continuous function \(f(x_1,x_2,x_3)\) of three variables on the unit cube \(E^3\) can be represented in the form

\[ f(x_1,x_2,x_3)=\sum_{i=1}^{3}\sum_{j=1}^{3} h_{ij}\,[\varphi_{ij}(x_1,x_2),x_3], \tag{1} \]

where the functions of two variables \(h_{ij}\) and \(\varphi_{ij}\) are real and continuous.

A. N. Kolmogorov recently obtained \((^1)\) the representation

\[ f(x_1,x_2,x_3)=\sum_{i=1}^{3} h_i\,[\varphi_i(x_1,x_2),x_3], \tag{2} \]

where the functions \(h_i\) and \(\varphi_i\) are continuous, the functions \(h_i\) are real, and the functions \(\varphi_i\) take values belonging to a certain tree \(\Xi\). The tree \(\Xi\) in A. N. Kolmogorov’s construction (for the case of a function of three variables) may be taken not universal, but such that all its points have branching index not exceeding 3. For this purpose the functions \(u_{km}^r\) of the basic lemma \((^1)\) (for \(n=2\)) should be chosen so that, in addition to the five properties indicated there, they also have the following properties:

6) The boundary of each level set of each function \(u_{km}^r\) divides the plane into no more than 3 parts.

7) For every \(r\), \(G_{11}^r \supset E^2\).

By virtue of this remark, Theorem 1 is a consequence of the existence of the representation (2) and of the following theorem:

Theorem 2. Whatever the family \(F\) of real equicontinuous functions \(f(\xi)\), defined on a tree \(\Xi\) all of whose points have branching index \(\leq 3\), one can realize the tree as a subset \(X\) of the three-dimensional cube \(E^3\) in such a way that any function of the family \(F\) can be represented in the form

\[ f(\xi)=\sum_{k=1}^{3} f_k(x_k), \]

where \(x=(x_1,x_2,x_3)\) is the image of \(\xi\in\Xi\) in the tree \(X\); \(f_k(x_k)\) are continuous real functions of one variable, with \(f_k\) depending continuously on \(f\) (in the sense of uniform convergence).

Let us introduce some auxiliary notions. Let \(K\) be a finite segmental complex situated in \(E^3\) and consisting of segments not parallel to any of the coordinate planes.

Definition 1. A system of points of \(K\)

\[ a_0 \ne a_1 \ne \cdots \ne a_{n-1} \ne a_n \]

is called a lightning, if the segments \(\overline{a_{i-1}a_i}\) are perpendicular, respectively, to the axes \(X_{\alpha_i}\) and

\[ \alpha_1 \ne \alpha_2 \ne \cdots \ne \alpha_{n-1} \ne \alpha_n . \]

A finite system of pairwise distinct points \(a_{i_1 i_2 \ldots i_n}\), numbered by tuples of indices \(i_1 i_2 \ldots i_n\), is called a branching scheme if: 1) there exists only one point \(a_0\), numbered by a single index; 2) together with \(a_{i_1 i_2 \ldots i_{n-1} i_n}\), the system contains \(a_{i_1 \ldots i_{n-1}}\).

Definition 2. A branching system of points \(a_{i_1 \ldots i_n}\) situated on \(K\) is called a deriving scheme if, for a fixed tuple \(i_1 \ldots i_n\), the totality of points of the form \(a_{i_1 \ldots i_n i_{n+1}}\) lies in the plane passing through \(a_{i_1 \ldots i_n}\), perpendicular to some coordinate axis \(x_{\alpha_{i_1 \ldots i_n}}\), and exhausts all points of intersection of this plane with \(K\) distinct from \(a_{i_1 \ldots i_n}\).

The tree \(\Xi\) can be represented in the form

\[ \Xi = \overline{\bigcup_{n=1}^{\infty} D_n}, \qquad D_n \subset D_{n+1}, \]

where \(D_n\) are finite trees, \(D_1\) is a simple arc, and \(D_{n+1}\) is obtained from \(D_n\) by attaching, at some point \(p_n\), which is for \(D_n\) neither a branching point nor an endpoint, the segment \(S_n\) \((^2)\).

Denote by \(\omega_n\) the upper bound of the oscillations of functions \(f \in F\) on the components of the difference \(\Xi \setminus D_n\). It is easy to see that

\[ \omega_n \to 0 \quad \text{as } n \to \infty . \]

Therefore one can choose a sequence

\[ n_1 < n_2 < \cdots < n_r < \cdots , \]

such that

\[ \omega_n \le \frac{1}{r^2} \quad \text{for } n \ge n_r . \]

A realization \(X\) of the tree \(\Xi\) in \(E^3\) is constructed in the form:

\[ X = \overline{\bigcup_{n=1}^{\infty} D'_n}, \]

where \(D'_n\) are segmental complexes realizing \(D_n\) in such a way that the images \(S'_n\) of the arcs \(S_n\) are segments not perpendicular to the coordinate axes.

The inductive construction of \(D'_n\) is carried out so that \(\overline{\bigcup_{n=1}^{\infty} D'_n}\) is a tree \((^2)\) and with the following conditions satisfied:

1) Every function \(f \in F\) is represented on \(D_n\) in the form

\[ f(\xi)=\sum_{k=1}^{3} f_k^n(x_k), \tag{3} \]

where \(f_k^n(x_k)\) depends continuously on \(f\).

2) The tree \(D_n'\) from any point \(a_0\) has an outgoing scheme in which the first direction \(\alpha_0\) may be chosen arbitrarily.

3) Let \(A_n\) be the set of points of \(D_n'\) that are images of branching points of \(E\). There exists a countable set \(B_n \subset D_n'\), \(B_n \cap A_n = 0\), such that broken lines \(a_0 \ldots a_m\), beginning at \(a_0 \in D_n' \setminus B_n\), have no common points with \(A_n\) and no coincident points \(a_i=a_j,\ i\ne j\).

4) If \(n_r<n\leqslant n_{r+1}\), then

\[ |f_k^n(x_k)-f_k^{n_r}(x_k)|\leqslant \left(3+\frac{n-n_r}{\,n_{r+1}-n_r\,}\right)\frac{1}{r^2}. \tag{4} \]

The proof of the possibility of the inductive construction of the trees \(D_n'\) and the functions \(f_k^n\), while preserving properties 1)—4), is too complicated to be presented here. Roughly speaking, at each step the attached segment \(S_{n+1}'\) is chosen very short; its direction and the manner of mapping \(S_{n+1}\) onto \(S_{n+1}'\) are chosen so as to ensure the fulfillment of properties 2) and 3) for \(D_{n+1}'\). Preservation of equality (3) in the transition from \(n\) to \(n+1\) on the newly attached segment \(S_{n+1}\) requires introducing a correction \(f_k^{n+1}-f_k^n\) to at least one of the functions \(f_k^n\) on the projection of \(S_{n+1}'\) onto the axis \(x_k\). In order to preserve equality (3) on the previously constructed tree \(D_n'\), this correction must be compensated by new corrections to the functions \(f_k^n\) on a number of other segments. We do not set out here the exact method of introducing these corrections. We note only the following: these corrections must be such that, for \(n'=n+1\), inequality (4) is preserved; if \(S_{n+1}'\) is sufficiently small and suitably placed, they can be made, for each function \(f_k^n\), on a finite system of pairwise nonintersecting segments of the axis \(x_k\); in the proof of this possibility, the circumstance that the tree \(D_n'\) has properties 2) and 3) is used essentially.

The proof of the existence of a continuous function

\[ f_k(x_k)=\lim_{n\to\infty} f_k^n(x_k) \]

and of the validity of the equality

\[ f(\xi)=\sum_{k=1}^{3} f_k(x_k) \]

on all of \(X\) is not difficult.

I am very grateful to A. N. Kolmogorov for his help and advice in carrying out this work.

Moscow State University
named after M. V. Lomonosov

Received
6 IV 1957

References

\(^{1}\) A. N. Kolmogorov, DAN, 108, No. 2, 179 (1956).
\(^{2}\) K. Menger, Kurventheorie, X, Berlin—Leipzig, 1932.