A. G. VITUSHKIN

PROOF OF THE EXISTENCE OF ANALYTIC FUNCTIONS OF SEVERAL VARIABLES NOT REPRESENTABLE BY LINEAR SUPERPOSITIONS OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS OF A SMALLER NUMBER OF VARIABLES

(Presented by Academician A. N. Kolmogorov on 30 III 1964)

Let \(G(\rho,z_1,z_2)\) be a domain in the space of two complex variables \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\), defined by the inequalities \(|y_i|\leq \rho\) \((i=1,2)\).

Theorem. For any functions \(p_m=p_m(x_1,x_2)\), continuous in the whole plane, and functions \(q_m=q_m(x_1,x_2)\) \((m=1,2,\ldots,N)\), continuously differentiable in the whole plane, and for any domain \(D\) of the plane of the variables \(x_1,x_2\), there exist a number \(\rho>0\) and a function \(f(z_1,z_2)\), analytic and bounded in the domain \(G(\rho,z_1,z_2)\), taking real values on the plane of the variables \(x_1,x_2\) \((y_1=y_2=0)\), and not equal in the domain \(D\) to any superposition of the form

\[ \sum_{m=1}^{N} p_m(x_1,x_2) f_m(q_m(x_1,x_2)), \]

where \(\{f_m(t)\}\) are arbitrary continuous functions.

It is already of interest to compare Theorem 1 with A. N. Kolmogorov’s theorem on the possibility of representing every continuous function of two variables by a superposition of the form

\[ \sum_{i=1}^{5} f_i(\alpha_i(x)+\beta_i(y)), \]

where all the functions are continuous, and \(\{\alpha_i(x)+\beta_i(y)\}\) are fixed in advance.

Notation: \(\omega(\delta)\) is the modulus of continuity of the functions \(\{p_m;\partial q_m/\partial x_1;\partial q_m/\partial x_2\}\); \(\operatorname{grad}[f(z)]\) is the gradient of \(f(z)\); \(d_1(e)\) is the one-dimensional diameter of the set \(e\); \(h_1(e)\) is the length of the set \(e\); \(c_1,c_2,\ldots\) are constants.

Lemma 1. In every domain \(D\) one can fix a closed subset \(G\), which is the union of a finite number of simply connected closed domains, specify a constant \(c>0\), and renumber the functions \(\{p_m;q_m\}\) by two indices in such a way that the newly obtained functions

\[ p_i^k=p_i^k(x_1,x_2), \qquad q_i^k=q_i^k(x_1,x_2) \]

\[ (i=0,1,2,\ldots,n;\ k=1,2,\ldots,m_i;\ \sum_{i=0}^{n} m_i\leq N), \]

i.e. some of the pairs of functions \(\{p_m;q_m\}\) under this renumbering may in general be omitted, satisfy the following eight conditions:

1. \(\{p_i^k;\partial q_i^k/\partial x_1;\partial q_i^k/\partial x_2\}\) have modulus of continuity \(\omega(\delta)\).

2. \(c\leq |p_i^k(x_1,x_2)|\leq c^{-1}\), \((x_1,x_2)\in G\).

3. For \(i=0\), \(q_i^k\equiv \mathrm{const}\) in \(G\), and for \(i>0\),
   \(c\leq |\operatorname{grad}[q_i^k(x_1,x_2)]|\leq c^{-1}\), \((x_1,x_2)\subset G\).

4. \(q_i^k(x_1,x_2)\equiv \varphi_i^{kl}(q_i^l(x_1,x_2))\), \((x_1,x_2)\in G\), where \(\varphi_i^{kl}(t)\) is a strictly monotone continuously differentiable function of \(t\).

5. If \(i\neq j\), then, for all \(k\) and \(l\), the absolute value of the acute angle formed by the level lines of the functions \(q_i^k\) and \(q_j^l\) passing through an arbitrary point \((x_1,x_2)\in G\) does not exceed \(c\).

6. The set \(G\) is the union of pairwise nonintersecting closed simply connected domains \(\{G_l\}\), each of which is such that,

for all \(i,k\) the intersection of the level set of the function \(q_i^k\) with this domain is either the empty set or a simple arc (with endpoints on the boundary \(G_l\)) of length not less than \(c\).

1. For every \(i>0\) and all functions \(\{f_i^k(t)\}\) the inequality holds
   \[ \sup_{(x_1,x_2)\in G}\left|\sum_{k=1}^{m_i}p_i^k(x_1,x_2)f_i^k(q_i^k(x_1,x_2))\right| \geq c\max_k\sup_{(x_1,x_2)\in G}\left|f_i^k(q_i^k(x_1,x_2))\right|. \]

2. For any bounded measurable functions \(\{\varphi_m(t)\}\) there exist measurable bounded functions \(\{f_i^k(t)\}\) such that
   \[ \sum_{i=0}^{n}\sum_{k=1}^{m_i}p_i^k f_i^k(q_i^k) = \sum_{m=1}^{N}p_m\varphi_m(q_m) \quad \text{in } G . \]

We shall prove Lemma 1 by induction on \(N\). For \(N=1\) the assertion of the lemma is easy to verify. Let \(N>1\). Fix a domain \(G^*\subset D\) and renumber the functions \(\{p_m;q_m\}\) by two indices so that the functions \(\{p_i^k;q_i^k\}\) in the domain \(G^*\) satisfy conditions 1)—5). Denote by \(e_{i,t}\) the level set \(t\) of the function \(q_i^1\); fix some set \(\Gamma\subset G^*\) and put
\[ \lambda_i(t,\Gamma) = \inf_{\{c_i^k\}} \sup_{(x_1,x_2)\in \Gamma\cap e_{i,t}} \left|\sum_{k=1}^{m_i}c_i^k p_i^k(x_1,x_2)\right|, \]
where the infimum is taken over all sets \(\{c_i^k\}\) such that \(\max_k |c_i^k|=1\).

The domain of definition of the function \(\lambda_i(t,\Gamma)\) is the set of values
\(t=q_i^1(x_1,x_2)\), \((x_1,x_2)\in \Gamma\). If \(\Gamma\) is closed, then \(\lambda_i(t,\Gamma)\) is continuous. If for \(\Gamma\) \(\lambda_i(t,\Gamma)=0\), then there exists a subdomain \(\Gamma^*\subset \Gamma\) and measurable functions \(c_i^k(t)\), bounded by one, such that
\[ \sum_{k=1}^{m_i}p_i^k(x_1,x_2)c_i^k(q_i^k(x_1,x_2))\equiv 0 \]
in \(\Gamma^*\), and for some \(k\)
\[ c_i^k(q_i^k(x_1,x_2))\equiv 1 \]
in \(\Gamma^*\). But this means that one of the terms of the superposition
\[ \sum_{k=1}^{m_i}p_i^k f_i^k(q_i^k) \]
can be excluded, without thereby narrowing the class of functions representable by superpositions of this kind (see condition 8). In this case Lemma 1 is proved by virtue of the corresponding induction hypothesis. Consequently, further we may assume that for every open set \(\Gamma\) the corresponding set \(\lambda_i(t,\Gamma)>0\) everywhere densely (and openly, by continuity of the function \(\lambda_i(t,\Gamma)\)) in \(q_i^1(\Gamma)\). If for the set \(\Gamma\)
\[ \lambda_i(t,\Gamma)\geq c_\Gamma=\mathrm{const}>0, \]
and \(\Gamma''\) is such that for every \(t\in q_i^1(\Gamma)\) the set \(e_{i,t}\cap \Gamma''\) is an \(\varepsilon\)-net in \(e_{i,t}\cap \Gamma\) (\(\varepsilon\) does not depend on \(t\) and is sufficiently small), then
\[ \lambda_i(t,\Gamma\cap \Gamma'')\geq \tfrac12 c_\Gamma . \]
From the last two assertions it follows that one can indicate open sets, consisting of a finite number of components,
\[ G^*\supset \Gamma_1\supset \Gamma_2\supset \cdots \supset \Gamma_n=\Gamma_{n+1}, \]
such that, for every \(i\),
\[ \lambda_i(t,\Gamma_{i+1})\geq c'=\mathrm{const}>0, \]
and each component of the set \(\Gamma_i\) has a boundary consisting of a finite number of segments of level lines of the functions \(\{q_j^1\}\). Then, for every \(i>0\) and all functions \(\{f_i^k(t)\}\),
\[ \sup_{(x_1,x_2)\in \Gamma_n} \left|\sum_{k=1}^{m_i}p_i^k(x_1,x_2)f_i^k(q_i^k(x_1,x_2))\right| \geq c'\max_k\sup_{(x_1,x_2)\in \Gamma_n} \left|f_i^k(q_i^k(x_1,x_2))\right|, \]
i.e., condition 7 is fulfilled.

We take for \(G\) the closure of the set \(\Gamma_n\). Fulfillment of condition 6 can be achieved already in defining the sets \(\Gamma_i\), by requiring that for every \(i\) the boundary of every component \(\gamma\) of the set \(\Gamma_i\) consist of a finite number of segments of level lines of the functions \(q_1^1,q_2^1,\ldots,q_n^1\) such that, if the arc \([a,b]\) passes into the arc \([b,d]\) \((q_i^1([a,b])=\)

\(= \mathrm{const}\) and \(q_m^1([b,d])=\mathrm{const}\)), then for every \(k \ne j,m\) the level line of the function \(q_k^1\) passing through the point \(b\) intersects the arc \([a,b,d]\), passing at the point \(b\) from the component \(\gamma\) to its complement. The lemma is proved.

Lemma 2. Let \([a',a'']\) and \([b',b'']\) be segments of level lines of the functions \(\{q_i^k\}\) (\(i\) fixed); \(\alpha=h_1([a',a''])\); \(d_1(a'\cup b')\le \delta\); \(d_1(a''\cup b'')\le \delta\). Then, if \(\{p_i^k\}\) and \(\{q_i^k\}\) satisfy conditions 1–6 and \(\delta\) is sufficiently small in comparison with \(\alpha\), then for all continuous functions \(\{f_i^k(t)\}\)

\[ \left| \int_{s=[a',a'']} \sum_{k=1}^{m_i} p_i^k(s) f_i^k(q_i^k(s))\,ds - \int_{s\in[b',b'']} \sum_{k=1}^{m_i} p_i^k(s) f_i^k(q_i^k(s))\,ds \right| \le \]

\[ \le c_1(\alpha\varepsilon+m\alpha\omega(\delta)+m\delta), \]

\[ \varepsilon=\max_{(x_1,x_2)\in G}\left|\sum_{i,k}p_i^k f_i^k(q_i^k)\right|; \qquad m=\max_{i,k}\max_{(x_1,x_2)\in G}|f_i^k(q_i^k)|, \]
\(c_1\) does not depend on \(\alpha,\delta,\varepsilon,m\).

Proof. On \([a',a'']\) fix a system of points \(a_1,a_2,\ldots,a_\nu\), uniformly distributed with respect to length (\(a'=a_1;\ a''=a_\nu\)), and denote by \(b_r\) the point of intersection of the level line of the function \(q_i^k\), containing the arc \([b',b'']\), with the level line of the function \(q_j^k\) passing through the point \(a_r\) (here \(j\ne i\) should be regarded as fixed). By Lemma 3, from (1) we have

\[ \left| \int_{s\in[a',a'']} p_j^k(s) f_j^k(q_j^k(s))\,ds - \int_{s\in[b',b'']} p_j^k(s) f_j^k(q_j^k(s))\,ds \right| = \]

\[ = \lim_{\nu\to\infty}\left| \sum_{r=1}^{\nu} p_j^k(a_r) f_j^k(q_j^k(a_r))\,h_1([a_r,a_{r+1}]) - \right. \]

\[ \left. - \sum_{r=1}^{\nu} p_j^k(a_r) f_j^k(q_j^k(a_r))\,h_1([a_r,a_{r+1}])(1+O(1)\omega(\delta)) \right| + \]

\[ +O(1)m(\delta+\alpha\omega(\delta)) = O(1)m(\delta+\alpha\omega(\delta)). \]

Then

\[ \left| \int_{s\in[a',a'']} \sum_{k=1}^{m_i} p_i^k(s) f_i^k(q_i^k(s))\,ds - \int_{s\in[b',b'']} \sum_{k=1}^{m_i} p_i^k(s) f_i^k(q_i^k(s))\,ds \right| \le \]

\[ \le c_2\varepsilon\alpha + n\max_{j\ne i,\ k}m_j \left| \int_{s\in[a',a'']} p_j^k(s) f_j^k(q_j^k(s))\,ds - \int_{s\in[b',b'']} p_j^k(s) f_j^k(q_j^k(s))\,ds \right| \le \]

\[ \le c_2\varepsilon\alpha+c_3m(\delta+\alpha\omega(\delta)) \le c_1(\alpha\varepsilon+m\delta+m\alpha\omega(\delta)). \]

The lemma is proved.

Let \(F=F(p,q,m,\varepsilon)\) be the set of superpositions of the form

\[ f(x_1,x_2)=\sum_{i=0}^{n}\sum_{k=1}^{m_i}p_i^k(x_1,x_2)\,f_i^k(q_i^k(x_1,x_2)) \]

such that
\[ \max_{(x_1,x_2)\in G}|f(x_1,x_2)|\le \varepsilon', \]
\(\{p_i^k\}\) and \(\{q_i^k\}\) satisfy conditions 1–8, and \(\{f_i^k\}\) are measurable and bounded by the constant \(m\). Put

\[ R(f(x_1,x_2),\delta) = \max_{S(\delta,x_1,x_2)} \left| \frac{1}{\pi\delta^2} \iint_{S(\delta,x_1,x_2)} f(u,v)\,du\,dv \right|, \]
where \(S(\delta,x_1,x_2)\) is the disk of radius \(\delta\) with center at the point \((x_1,x_2)\). Denote by \(\mathcal H_{\varepsilon^*}^{\delta}(F)\) the \(\varepsilon^*\)-entropy of the space \(F\), taking as the distance between functions \(f_1(x_1,x_2)\) and \(f_2(x_1,x_2)\in F\) the number
\[ R(f_1(x_1,x_2)-f_2(x_1,x_2),\delta). \]

Lemma 3. If \(0<\theta\le 1\) and \(m/\theta\varepsilon\ge 2\), then

\[ \mathcal H_{\theta\varepsilon}^{\delta}(F(p,q,m,\varepsilon)) \le c_4(1/\theta^2\delta+\log m/\varepsilon), \]

where \(c_4\) does not depend on \(m,\varepsilon,\theta,\delta\); \(\theta>A\omega(\delta)\).

Proof. Let \(e_{i,j}\) be level sets of the functions \(q_i'\) \((i=1,\ldots,n;\ j=1,2,\ldots,r_i)\) such that, for every \(i\),
\[ e_i=\bigcup_{j=1}^{r_i} e_{ij} \]
is a \(\beta\)-net in \(G\) (\(\beta\) will be fixed below). We partition the set \(F\) into the smallest possible number of subsets \(F_1,\ldots,F_r\) such that, for every \(\nu\) and for any functions \(f_1(x_1,x_2)\) and \(f_2(x_1,x_2)\) from \(F_\nu\), their difference
\[ \widetilde f(x_1,x_2)=f_1(x_1,x_2)-f_2(x_1,x_2) =\sum_{i,k} p_i^k f_i^k(q_i^k) \]
is such that, for all \(i,k,j\),
\[ \left|\widetilde f_i^k\bigl(q_i^k(e_{i,j})\bigr)\right|\le c_5\varepsilon . \]
For fixed \(c_5\) and \(\beta\),
\[ r\le \frac{c_6 m}{\varepsilon}. \]
We shall show that
\[ \mu=\max_{i,k}\max_{(x_1,x_2)\in G} \left|\widetilde f_i^k\bigl(q_i^k(x_1,x_2)\bigr)\right|\le c_7\varepsilon . \]
Suppose, for definiteness, that
\[ \widetilde f_1^1(q_1^1(a))=m_1;\qquad a\in G. \]
By condition 7, at some point \(a'\in G\),
\[ |\varphi_1(a')| = \left|\sum_{k=1}^{m_1} p_1^k(a')\, \widetilde f_1^k\bigl(q_1^k(a')\bigr)\right| \ge c\mu . \]

Let \([a',a'']\subset G\) be a segment of a level line of the function \(q_1^k\) such that
\[ \omega(\alpha)=\omega\bigl(h_1[a',a'']\bigr)\le c[2m_1]^{-1}. \]
On \([a',a'']\), \(\varphi_1(x_1,x_2)\) preserves a constant sign, and for \((x_1,x_2)\in [a',a'']\),
\[ |\varphi_1(x_1,x_2)|\ge \tfrac12 c\mu . \]
Consequently,
\[ \left|\int_{s\in[a',a'']} \varphi_1(s)\,ds\right| \ge \tfrac12 c\mu\alpha . \]

Let \([b',b'']\) be a segment of one of the lines \(\{e_{i,j}\}\) such that
\[ d_1(a'\cup b')\le 3\beta \quad\text{and}\quad d_1(a''\cup b'')\le 3\beta . \]
By the definition of \(F_1\) it follows that
\[ \left|\int_{s\in[b',b'']} \varphi_1(s)\,ds\right| \le c_8\varepsilon\alpha', \]
where
\[ \alpha'=h_1([b',b'']). \]
Hence
\[ \left|\int_{s\in[a',a'']} \varphi_1(s)\,ds - \int_{s\in[b',b'']} \varphi_1(s)\,ds\right| \ge \tfrac12 c\mu\alpha-c_8\varepsilon\alpha' . \]
Assuming that \(\beta\) is sufficiently small in comparison with \(\alpha\), we obtain \(\alpha\sim\alpha'\), and from Lemma 2
\[ \left|\int_{s\in[a',a'']} \varphi_1(s)\,ds - \int_{s\in[b',b'']} \varphi_1(s)\,ds\right| \le c_1\bigl(\alpha\varepsilon+\mu\alpha\omega(3\beta)+3\mu\beta\bigr). \]
Thus,
\[ c_1\bigl(\varepsilon\alpha+\alpha\mu\omega(3\beta)+3\mu\beta\bigr) \ge \tfrac12 c\mu\alpha-c_8\varepsilon\alpha'. \]
From the last inequality we obtain \(\mu\le c_7\varepsilon\), and then, from Theorem 2 \((^1)\), we obtain
\[ \mathcal H_{\theta\varepsilon}^{\delta}(F_\nu) \le \frac{c_9}{\delta}\left(\frac{M}{\theta\varepsilon}\right)^2 . \]
Consequently,
\[ \mathcal H_{\theta\varepsilon}^{\delta}(F) \le \left(\frac{\mu}{\theta\varepsilon}\right)^2\frac{c_9}{\delta} +\log r \le c_4\left(\frac{1}{\theta^2\delta}+\log\frac{m}{\varepsilon}\right). \]
The lemma is proved.

Denote by \(\Phi(\rho,\mu,\varepsilon)\) the set of functions \(f(z_1,z_2)\) analytic in the domain \(G(\rho,z_1,z_2)\), bounded in \(G\) by the constant \(\mu>0\), real in the plane \(y_1=y_2=0\), and bounded on this plane by the constant \(\varepsilon>0\).

Lemma 4. For any positive numbers \(\rho,\mu\), \(\delta\le \delta_0(G)\) (\(G\) as in Lemma 1),
\[ \varepsilon=e^{-(\rho/\delta)^2}\mu \]
and for some \(\theta=\theta(\rho,\mu)\) (\(\theta\) does not depend on \(\delta\)), the inequality
\[ \mathcal H_{\theta\varepsilon}^{\delta}\bigl(\Phi(\rho,\mu,\varepsilon)\bigr) \ge \delta^{-2}c(G)\operatorname{mes}_2(G) \]
holds, where \(c(G)\) is a constant determined by the domain \(G\).

Using Lemmas 3 and 4, and assuming that \(\rho\) is sufficiently small, it is no longer difficult to prove the theorem with the aid of Lemma 1.

Received
30 III 1964

CITED LITERATURE

1. A. G. Vitushkin, DAN, 156, No. 5 (1964).