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) is a tree ((^2)) and with the following conditions satisfied:}^{\infty} D'_n

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