MATHEMATICS

A. G. VITUSHKIN

SOME PROPERTIES OF LINEAR SUPERPOSITIONS OF SMOOTH FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 28 III 1964)

§ 1. The “dimension” of a space of functions of \(n\) real variables. Let \(G_n\) be a domain of \(n\)-dimensional Euclidean space; \(C(G_n)\) the space of all continuous real functions in \(G_n\). Functions \(f_1(x), f_2(x) \in C(G_n)\) will be called \((\varepsilon,\delta)\)-distinguishable if there exists an \(n\)-dimensional ball \(S_\delta \subset G_n\) of radius \(\delta\) such that
\[ \min_{x\in S_\delta}|f_1(x)-f_2(x)|\geq \varepsilon . \]

Denote by \(N_{\varepsilon,\delta}(E)\) the number of elements of a maximal subset of \(E \subset C(G_n)\), any two elements of which are \((\varepsilon,\delta)\)-distinguishable.

Theorem 1. Let a set \(F \subset C(G_n)\), everywhere dense in the uniform metric, together with each element \(f\), contain the element \(\lambda f\); \(\rho(f)\) is a nonnegative functional, defined on \(F\), such that for every \(f\in F\)
\(\rho(\lambda f)\to 0\) as \(\lambda\to 0\); \(F_{\varepsilon,r}\) is the intersection of two sets
\[ \{f:\max_{x\in G}|f(x)|\leq \varepsilon;\ f(x)\in C(G_n)\} \quad\text{and}\quad \{f:\rho(f)\leq r;\ f\in F\}. \]
Then, for every \(r>0\),
\[ \lim_{\delta\to0}\lim_{\varepsilon\to0} \left[-\,\frac{\log\log N_{\varepsilon,\delta}(F_{\varepsilon,r})}{\log\delta}\right]=n . \]

From Theorem 1 we obtain that, for example, the space of analytic functions of \(n\) real variables and the space of all continuous functions of \(n\) real variables have the same dimension, equal to \(n\). Relying on this property of the space of functions of \(n\) variables, we shall prove, for example, the existence of analytic functions of \(n\) variables not representable by superpositions of the form
\[ \sum_{k=1}^{\nu} p_k f_k\bigl(q_k^1,q_k^2,\ldots,q_k^{\,n-1}\bigr), \]
where \(\{f_k\}\) are arbitrary continuous functions of \(n-1\) variables, and
\(\{q_k^i=q_k^i(x_1,x_2,\ldots,x_n)\}\) are arbitrary previously fixed continuously differentiable functions of \(n\) variables; \(p_k\) are fixed continuous functions of \(n\) variables. For simplicity of exposition we shall restrict ourselves to the consideration of functions of only one and two variables.

§ 2. Level lines of smooth functions.
Notation. \(z\) is an arbitrary point of the two-dimensional plane with coordinates \(x,y\); \(\operatorname{grad}[f(z)]\) is the gradient of the function \(f(z)\); \(e(f,t)\) is the level set \(t\) of the function \(f=f(z)\); \(\vec{\tau}(e,z)\) is the unit tangent vector to the line \(e\) at the point \(z\in e\); \(\gamma(\vec{\tau}_1,\vec{\tau}_2)\) is the absolute value of the acute angle between the vectors \(\vec{\tau}_1,\vec{\tau}_2\); \(d_1(e)\) is the one-dimensional diameter of the set \(e\); \(h_1(e)\) is the length of the set \(e\); \(h_2(e)\) is the area of the set \(e\); \(S(\delta,z)\) is the circle of radius \(\delta\) with center at the point \(z\); \(c_k\) \((k=1,2,\ldots)\) are constants.

Let \(p=p(z)\) and \(q=q(z)\) be functions defined in a simply connected closed domain \(G\) and possessing the following properties: a) \(p(z)\) is continuous in \(G\) and has modulus of continuity \(\omega(\delta)\), and \(|p(z)|\leq M<\infty\); b) \(\partial q/\partial x\) and \(\partial q/\partial y\) have in the domain \(G\) the same modulus of continuity \(\omega(\delta)\), and \(0<m'\leq\)

\[ \leqslant |\operatorname{grad}[q(z)]|\leqslant M'<\infty. \]
The constants \(\{c_k\}\) will henceforth be assumed to depend only on \(M, m', M', d_1(G), \omega(d_1(G))\).

Lemma 1. Let \(z_1\in e_1=e(q_1,t_1)\) and \(z_2\in e_2=e(q_1,t_2)\). Then

\[ \gamma\bigl(\vec\tau(e_1,z_1),\vec\tau(e_2,z_2)\bigr)\leqslant c_1\omega(\delta), \qquad \text{where } \delta=d_1(z_1\cup z_2). \]

Lemma 2. If \(\beta(e)\leqslant \pi/4\) is the magnitude of the angle swept out by the tangent vector to the connected line \(e\), then
\[ h_1(e)\leqslant d_1(e)(1+c_2\beta(e)). \]

Lemma 3. Let \(q_1=q_1(z)\) and \(q_2=q_2(z)\) be functions defined in the domain \(G\), possessing property b), and such that for every \(z\in G\)
\[ \gamma\{\vec\tau[e(q_2,q_2(z)),z],\vec\tau[e(q_1,q_1(z)),z]\}\geqslant \gamma_0>0 \qquad (\gamma_0=\text{const}). \]
Let \(e'_{q_2}\) and \(e''_{q_2}\) be two level lines of the function \(q_2\), and \(e'_{q_1}\) and \(e''_{q_1}\) level lines of the function \(q_1\); let \([a',a'']\) be a segment of the line \(e'_{q_1}\) with endpoints \(a'\in e'_{q_2}\) and \(a''\in e''_{q_2}\); let \([b',b'']\) be a segment of the line \(e''_{q_1}\) with endpoints \(b'\in e'_{q_2}\) and \(b''\in e''_{q_2}\).

Then
\[ h_1([b',b''])\leqslant h_1([a',a''])\bigl(1+c_3\omega(\delta)/\gamma_0\bigr), \]
where
\[ \delta=d_1([a',a'']\cup [b',b'']). \]

Proof. Since
\[ q_2(a'')-q_2(a')=q_2(b'')-q_2(b'), \]
we have
\[ \int_{[a',a'']} \frac{\partial q_2}{\partial s}\,ds = \int_{[b',b'']} \frac{\partial q_2}{\partial s}\,ds. \]
Consequently,
\[ \frac{\partial q_2(a^*)}{\partial s}\,h_1([a',a'']) = \frac{\partial q_2(b^*)}{\partial s}\,h_1([b'_1,b''_2]), \]
where \(\dfrac{\partial q_2(a^*)}{\partial s}\) and \(\dfrac{\partial q_2(b^*)}{\partial s}\) are the derivatives at the points \(a^*\in [a',a'']\) and \(b^*\in [b',b'']\) along the lines \([a',a'']\) and \([b',b'']\), respectively. Denote by \(q_2^*\) the derivative of \(q_2\) at the point \(b^*\) along the direction \(\vec\tau(e'_{q_1},a^*)\), and put
\[ \alpha=\gamma\{\vec\tau[e''_{q_1},b^*],\vec\tau[e'_{q_1},a^*]\}. \]
By Lemma 1,
\[ \alpha\leqslant c_1\omega(\delta), \]
where
\[ \delta=d_1([a',a'']\cup [b',b'']), \]
and therefore
\[ \frac{\partial q_2(a^*)}{\partial s} = q_2^*+O(1)\omega(\delta) = \frac{\partial q_2(b^*)}{\partial s} + O(1)\left\{\left|q_2^*-\frac{\partial q_2(b^*)}{\partial s}\right|+\omega(\delta)\right\} = \]
\[ = \frac{\partial q_2(b^*)}{\partial s} + O(1)\{\alpha+\omega(\delta)\} = \frac{\partial q_2(b^*)}{\partial s} + O(1)\omega(\delta). \]

Consequently,
\[ h_1([b',b'']) = h_1([a',a'']) \frac{\partial q_2(a^*)}{\partial s} \left(\frac{\partial q_2(b^*)}{\partial s}\right)^{-1} = h_1([a',a''])\times \]
\[ \times \left(1+O(1)\omega(\delta) \left(\frac{\partial q_2(b^*)}{\partial s}\right)^{-1}\right) = h_1([a',a'']) \left(1+O(1)\frac{\omega(\delta)}{\gamma_0}\right), \]
since
\[ \left|\frac{\partial q_2(b^*)}{\partial s}\right| \geqslant |\operatorname{grad}[q_2(b^*)]|\sin\gamma_0. \]
The lemma is proved.

Lemma 4. Let \(S(\delta,z)\subset G\); let \(\mu_q(t)\) be the function equal to
\[ \bigl[\delta^2-(t-q(z))^2|\operatorname{grad}[q(z)]|^{-2}\bigr]^{1/2} \]
on the interval
\[ q(z)-\delta|\operatorname{grad}[q(z)]|\leqslant t\leqslant q(z)+\delta|\operatorname{grad}[q(z)]| \]
and equal to zero outside this interval. Then
\[ \int_{-\infty}^{\infty} \left|\mu_q(t)-h_1(e(q,t)\cap S(\delta,z))\right|\,dt \leqslant c_4\omega(\delta)\delta^2. \]

Proof. Let \([a,b]\subset e(q,t)\cap S(\delta,z)\) be a segment of a level line with endpoints \(a\) and \(b\) lying on the boundary of \(S(\delta,z)\); let
\[ \alpha_1=\gamma((\overline{za},\operatorname{grad}[q(z)])),\qquad \alpha_2=\gamma((\overline{zb},\operatorname{grad}[q(z)])). \]
We have
\[ |t-q(z)|=|q(a)-q(z)| = \left|\int_{[z,a]}\frac{\partial q}{\partial s}\,ds\right| = \]
\[ = \delta\cos\alpha_1\,|\operatorname{grad}[q(z)]|\,(1+O(1)\omega(\delta)). \]
Consequently,
\[ \delta\sin\alpha_1 = \bigl[\delta^2-(t-q(z)+O(1)\delta\omega(\delta))^2|\operatorname{grad}[q(z)]|^{-2}\bigr]^{1/2}. \]
Similarly,
\[ \delta\sin\alpha_2 = \bigl[\delta^2-(t-q(z)+O(1)\delta\omega(\delta))^2|\operatorname{grad}[q(z)]|^{-2}\bigr]^{1/2}. \]

If \(\alpha_1 \ge c_5\omega(\delta)\) (\(c_5\) is a sufficiently large constant), then \([a,b]=e(q,t)\cap S(\delta,z)\). By Lemmas 1 and 2,

\[ h_1[a,b]=\delta(\sin\alpha_1+\sin\alpha_2)(1+O(1)\omega(\delta))= \]

\[ =2\{\delta^2-(t-q(z)+O(1)\delta\omega(\delta))^2|\operatorname{grad}[q(z)]|^{-2}\}^{1/2} +O(1)\delta\omega(\delta), \]

and since for every \(t\)

\[ h_1(e(q,t)\cap S(\delta,z))\le c_6\delta[1+\omega(\delta)], \]

we have

\[ \int_{-\infty}^{\infty}\left|h_1(e(q,t)\cap S(\delta,z))-\mu_q(t)\right|\,dt= \]

\[ =\int_{q(z)-\theta}^{q(z)+\theta} \left|h_1(e(q,t)\cap S(\delta,z))-\mu_q(t)\right|\,dt +O(1)\delta^2\omega(\delta), \]

where \(\theta=\delta\cos[c_5\omega(\delta)]|\operatorname{grad}[q(z)]|\). Further,

\[ \int_{q(z)-\theta}^{q(z)+\theta} \left|h_1(e(q,t)\cap S(\delta,z))-\mu_q(t)\right|\,dt = O(1)\delta^2\omega(\delta)\int_{-1}^{1}\frac{dt}{\sqrt{1-t^2}}, \]

since

\[ h_1(e(q,t)\cap S(\delta,z))=h_1([a,b]) =2\{\delta^2-(t-q(z)+O(1)\delta\omega(\delta))^2|\operatorname{grad}[q(z)]|^{-2}\}^{1/2} +O(1)\delta\omega(\delta). \]

The lemma is proved.

Lemma 5. Let \(p(z), q(z)\) satisfy conditions a), b); \(S(\delta,z)\subset G\); and let \(f(t)\) be an arbitrary continuous function uniformly bounded in absolute value by a constant \(m\). Then

\[ \iint_{S(\delta,z)} p(u,v)f(q(u,v))\,du\,dv = p(z)|\operatorname{grad}[q(z)]|^{-1}\int_{-\infty}^{\infty} f(t)\mu_q(t)\,dt + \lambda(z)m\delta^2\omega(\delta), \]

where \(|\lambda(z)|\le c_7\).

Proof.

\[ \iint_{S(\delta,z)} p(u,v)f(q(u,v))\,du\,dv = p(z)\iint_{S(\delta,z)} f(q(u,v))\,du\,dv+ \]

\[ +O(1)m\delta^2\omega(\delta) = p(z)\int_{-\infty}^{\infty} \left\{ f(t)\int_{e(q,t)\cap S(\delta,z)} |\operatorname{grad}[q(u,v)]|^{-1}\sqrt{(du)^2+(dv)^2} \right\}dt+ \]

\[ +O(1)m\delta^2\omega(\delta) = p(z)|\operatorname{grad}[q(z)]|^{-1}\times \]

\[ \times\int_{-\infty}^{\infty} \left\{ f(t)\int_{e(q,t)\cap S(\delta,z)} \sqrt{(du)^2+(dv)^2} \right\}dt + O(1)m\delta^2\omega(\delta) = \]

\[ = p(z)|\operatorname{grad}[q(z)]|^{-1} \int_{-\infty}^{\infty} f(t)h_1(e(q,t)\cap S(\delta,z))\,dt + O(1)m\delta^2\omega(\delta) = \]

\[ = p(z)|\operatorname{grad}[q(z)]|^{-1} \int_{-\infty}^{\infty} f(t)\mu_q(t)\,dt + O(1)m\delta^2\omega(\delta) \]

(see Lemma 4). The lemma is proved.

Lemma 6. Let a number \(\alpha>0\) and functions \(p(z), q(z), f(t)\), satisfying the conditions of Lemma 5, be given. Then, if for every integer

\[ k\left\{\min_{z\in G}q(z)\le t_k=k\delta\frac{\alpha}{m}\le \max_{z\in G}q(z)\right\} \]

and every integer

\[ l\left\{\min_{z\in G}|\operatorname{grad}[q(z)]|\le t'_l=l\frac{\alpha}{m}\le \max_{z\in G}|\operatorname{grad}[q(z)]|\right\} \]

the inequality

\[ \left| \int_{t_k-t'_l\delta}^{t_k+t'_l\delta} f(t)\sqrt{\delta^2-\left(\frac{t-t_k}{t'_l}\right)^2}\,dt \right| \le \alpha\delta^2, \]

then for every disk \(S(\delta,z)\in G\)

\[ \left|\iint_{S(\delta,z)} p(u,v) f(q(u,v))\,du\,dv\right| \leq c_8\bigl(a\delta^2+m\delta^2\omega(\delta)\bigr). \]

The proof is easily obtained from Lemma 5. Denote by
\(F=F(m,p_1,p_2,\ldots,p_n,q_1,q_2,\ldots,q_n)\) the set of functions of the form
\(f(z)=\sum_{i=1}^{n} p_i(z) f_i(q_i(z))\), where \(\{p_i\}, \{q_i\}\) are fixed functions satisfying conditions a), b) (the constants \(M,m',M'\) do not depend on \(i\)), and the \(f_i(t)\) are arbitrary measurable functions uniformly bounded in modulus by the constant \(m\).

Put

\[ R(f(z),\delta)=\max_{S(\delta,z)\in G} \left|\frac{1}{\pi\delta^2}\iint_{S(\delta,z)} f(u,v)\,du\,dv\right|. \]

Denote by \(\mathcal H_\varepsilon^\delta(F)\) the \(\varepsilon\)-entropy of the space \(F\), taking as the distance between functions \(f_1(z), f_2(z)\in F\) the number \(R(f_1(z)-f_2(z),\delta)\).

Theorem 2. There exist constants \(A\) and \(B\) such that, if \(\varepsilon\geq A m\omega(\delta)\), then

\[ \mathcal H_\varepsilon^\delta(F)\leq \frac{B}{\delta}\left(\frac{m}{\varepsilon}\right)^2, \]

where \(A\) and \(B\) do not depend on \(m,\varepsilon,\delta\).

Proof. Without loss of generality, we may assume that the functions \(\{f_i(t)\}\) are continuous and are equal to zero outside the intervals \(\{[a_i,b_i]\}\), where
\(a_i=\min_{z\in G} q_i(z)\), \(b_i=\max_{z\in G} q_i(z)\). In order to compute

\[ f_\delta(z)=\frac{1}{\pi\delta^2}\iint_{S(\delta,z)\in G} f(u,v)\,du\,dv \]

to accuracy \(\varepsilon\), it is sufficient to specify the values

\[ v_i(t_k,t_l')= \frac{1}{\pi\delta^2} \int_{t_k-t_l'\delta}^{t_k+t_l'\delta} f_i(t)\sqrt{\delta^2-\left(\frac{t-t_k}{t_l'}\right)^2}\,dt \]

to accuracy \(\alpha=\pi\varepsilon/2nc_8\), and to assume \(\delta\) so small that
\(\varepsilon\geq 2nc_8m\omega(\delta)/\pi=A m\omega(\delta)\) (see Lemma 6). To record the numbers
\(v_i(t_k,t_l')\) (\(i,l\) fixed) to accuracy \(\alpha\), it is sufficient to have
\(N_{i,l}=c_9[\log m/\alpha+(b_i-a_i)m/\delta\alpha]\) binary digits. Here it should be taken into account that \(v_i(t_k,t_l')\) and \(v_i(t_{k+1},t_l')\) are sufficiently close; therefore, for storing
\(v_i(t_{k+1},t_l')-v_i(t_k,t_l')\) it is sufficient to have a number of digits not depending on \(m/\alpha\). Consequently, the total number of digits sufficient for recording the function \(f_\delta(z)\) is

\[ N=\sum_{i,l} N_{i,l} \leq B(n)c_{10}\left(\log\frac{m}{\alpha}+\frac{m}{\delta\alpha}\right) \frac{m}{\alpha}(M'-m') \leq \frac{B}{\delta}\left(\frac{m}{\varepsilon}\right)^2. \]

Consequently,

\[ \mathcal H_\varepsilon^\delta(F)\leq \frac{B}{\delta}\left(\frac{m}{\varepsilon}\right)^2. \]

The theorem is proved.

Corollary. If the functions \(p_1,p_2,\ldots,p_n\) continuous in the whole plane, the functions \(q_1,q_2,\ldots,q_n\) continuously differentiable, and a number \(M>0\) are fixed, then for every closed region \(G\) the set of superpositions of the form
\[ \sum_{k=1}^{n} p_k f_k(q_k), \]
where \(\{f_k\}\) are arbitrary continuous functions uniformly bounded by the constant \(M\), is nowhere dense in the space of all functions continuous on \(G\).

Received
28 III 1964