UDC 513.881

MATHEMATICS

A. F. TIMAN

ON THE \(\varepsilon\)-ENTROPY OF ARZELÀ COMPACTA OF FUNCTIONS DEFINED ON CLOSED SETS OF POSITIVE LEBESGUE MEASURE

(Presented by Academician A. N. Kolmogorov on 27 I 1967)

Let \(A\) be an arbitrary bounded closed set of \(n\)-dimensional Euclidean space \(R_n\). Any nondecreasing continuous and subadditive function \(\omega(t)\), defined on the half-axis \(0 \le t < \infty\), determines the class \(D_\omega^A(C)\) of all real functions
\(f(x)=f(x_1,\ldots,x_n)\), given at the points \(x \in A\), which satisfy the condition

\[ |f(x')-f(x'')| \le \omega(|x'-x''|) \qquad (x',x'' \in A) \]

and the condition

\[ |f(x)| \le C \qquad (x \in A). \]

The equality \(\lim_{t\to 0}\omega(t)=0\), by Arzelà’s theorem, is equivalent to the compactness of \(D_\omega^A(C)\) in the uniform metric

\[ \rho(f,\varphi)=\max_{x\in A}|f(x)-\varphi(x)|, \]

and the rate at which \(\omega(t)\) decreases to zero as \(t\to 0\) in general characterizes the measure of massiveness of the compactum \(D_\omega^A(C)\) in this theorem.

In the well-known work of A. N. Kolmogorov \((^1)\), in connection with the problem of superpositions of functions having prescribed smoothness (see \((^2)\)), and in the spirit of the ideas of information theory, as a measure of the massiveness of a compactum \(Q\) in a metric space \(R\), there were introduced its absolute \(\varepsilon\)-entropy \(H_\varepsilon(Q)\), the \(\varepsilon\)-entropy \(H_\varepsilon^R(Q)\) of the set \(Q\) relative to \(R\), and the \(\varepsilon\)-capacity \(\mathcal E_\varepsilon(Q)\) (see \((^3)\), § 1). In the same article \((^1)\) the problem was posed of investigating the growth of these characteristics as \(\varepsilon \to 0\) for various compacta \(Q\) occurring in analysis; the inequality

\[ \mathcal E_{\varepsilon 2}(Q) \le H_\varepsilon(Q) \le H_\varepsilon^R(Q) \le \mathcal E_\varepsilon(Q), \]

was given, its role in such an investigation was shown, and exact orders of growth of the \(\varepsilon\)-entropy (\(\varepsilon\)-capacity) were obtained for certain important cases.

One of the first results of the corresponding table of A. N. Kolmogorov (see \((^1)\), estimate III, 2) consists in the following: if \(A\) is some \(n\)-dimensional cube in \(R_n\), and \(\omega(t)=t^\alpha\) \((0<\alpha\le 1)\), then the following relation of order holds*:

\[ \mathcal E_\varepsilon\{D_\omega^A(C)\} \asymp H_\varepsilon\{D_\omega^A(C)\} \asymp \varepsilon^{-n/\alpha} \qquad (\omega(t)=t^\alpha). \tag{2} \]

Further investigations, connected with the attempt to generalize this result to more massive functional compacta \(D_\omega^A(C)\) arising when one considers Arzelà characteristics \(\omega(t)\) that decrease slowly to zero, or other compact-in-themselves sets \(A\) (see \((^4)\), § 17; \((^3)\), § 9; \((^5)\), theorem 8; \((^{11})\), pp. 3, 27), even for the simplest compacta \(A \subset R_n\) (an interval, a rectangle, a parallelepiped), left open

* Here and below we use the notation adopted by N. Bourbaki in \((^9)\), Chap. 5, for comparing infinitely small and infinitely large quantities.

the question of the exact order of growth of the $\varepsilon$-entropy $H_\varepsilon\{D_\omega^A(C)\}$ and the $\varepsilon$-capacity $\mathfrak E_\varepsilon\{D_\omega^A(C)\}$ in these cases, reducing only to estimates of the form (see ($^4$), § 17)

\[ 2^{H_{2\omega^{-1}(2\varepsilon)}(A)} \preccurlyeq \mathfrak E_{2\varepsilon}\{D_\omega^A(C)\} \preccurlyeq H_\varepsilon\{D_\omega^A(C)\} \preccurlyeq 2^{H_{1/2\omega^{-1}(\varepsilon/2)}(A)} \tag{3} \]

under the assumption that $A$ is connected, and to estimates of the form (see ($^3$), § 9, p. 77)

\[ 2^{H_{2\omega^{-1}(2\varepsilon)}(A)} \preccurlyeq \mathfrak E_{2\varepsilon}\{D_\omega^A(C)\} \preccurlyeq H_\varepsilon\{D_\omega^A(C)\} \preccurlyeq 2^{H_{1/2\omega^{-1}(\varepsilon/2)}(A)} \log \frac{1}{\varepsilon} \tag{4} \]

without this assumption.

The estimate (3), due to A. G. Vitushkin ($^4$), makes it possible to obtain the order of growth of the $\varepsilon$-entropy $H_\varepsilon\{D_\omega^A(C)\}$ for connected compact sets $A \subset R_n$, generally speaking, only when $\omega^{-1}(2\varepsilon) \asymp \omega^{-1}(\varepsilon)$. Even in the case where $A$ is a finite interval of the real axis, it is not difficult to give examples showing that without this additional restriction on the massiveness of $D_\omega^A(C)$ the extreme terms of inequality (3) may turn out to be infinitely large quantities of different orders.

In connection with the problem under consideration, the author ($^7$) established the following

Theorem 1. If a connected compact set $A$ in a metric space $R$ satisfies the condition

\[ 2^{H_\varepsilon(A)} \asymp 2^{H_\tau(A)} \tag{5} \]

for $\varepsilon \asymp \tau$, then always

\[ 0<\varliminf_{\varepsilon\to0} H_\varepsilon\{D_\omega^A(C)\}\cdot 2^{-H_{\omega^{-1}(2\varepsilon)}(A)} <\infty, \tag{6} \]

\[ \varlimsup_{\varepsilon\to0} H_\varepsilon\{D_\omega^A(C)\}\cdot 2^{-H_{\omega^{-1}(2\varepsilon)}(A)} <\infty, \tag{7} \]

where

\[ \omega(t_1)+\omega(t_2)\leq 2\omega\!\left(\frac{t_1+t_2}{2}\right). \tag{8} \]

In particular, if the closed bounded set $A$ belonging to the space $R_n$ is connected and

\[ H_\varepsilon(A)=n\log_2 1/\varepsilon+O(1) \tag{9} \]

(for example, if $A$ is an $n$-dimensional parallelepiped), then, when condition (8) is satisfied, the relation of exact order is always valid (see ($^6$))

\[ \mathfrak E_{2\varepsilon}\{D_\omega^A(C)\} \asymp H_\varepsilon\{D_\omega^A(C)\} \asymp \left\{\frac{1}{\omega^{-1}(2\varepsilon)}\right\}^{n}, \tag{10} \]

showing that the order in the right-hand estimate (3) is crude if $\omega(t)$ tends to zero sufficiently slowly.

The last result, giving exact orders of the $\varepsilon$-entropy and $\varepsilon$-capacity of arbitrarily massive Arzelà compacta of real functions $f(x_1,\ldots,x_n)$ of $n$ real variables and revealing an essential difference in their growth as $\varepsilon\to0$, naturally leads to the question of what the sets $A\subset R_n$ are for which (10) always holds. For such sufficiently massive compacta $Q=D_\omega^A(C)$, in the orders of growth of the corresponding terms of inequality (1) as $\varepsilon\to0$ there appear arbitrarily large gaps, whose absence in other cases plays the decisive role in obtaining the known estimates (see ($^3$)).

Theorem 2. For relation (10) to hold it is necessary, and when condition (8) is satisfied it is sufficient, that the compact set $A\subset R_n$ have positive $n$-dimensional Lebesgue measure.

The proof of this assertion is based on Theorem 1, the consideration of Hausdorff $p$-measures ($^{10}$), and the extension of functions with preservation of their modulus of continuity and maximum modulus.

Theorem 2 shows that the requirement that the set \(A \subset R_n\) be connected is not dictated by the nature of the question. In particular, the result of A. N. Kolmogorov \((^2)\), for any natural \(n\), remains valid for all closed sets \(A \subset R_n\) having positive \(n\)-dimensional Lebesgue measure, and only for them. For \(n=1\) this conclusion also follows from the estimate recently obtained by Vosburg \((^8)\) for one-dimensional compacta \(A\) and \(\omega(t)=t^\alpha\) \((0<\alpha \leq 1)\),

\[ H_\varepsilon\{D_{t^\alpha}^{A}(1)\}\preccurlyeq N_\delta(A)\log\{2\varepsilon^{-1}[N_\delta(A)]^{-\alpha}\}+\log \frac{1}{\varepsilon}, \]

where \(N_\delta(A)=2^{H_\delta(A)}\), \(\delta=\varepsilon^{1/\alpha}\), if one uses the particular case \(n=1\) (see \((^8)\)) of the following general proposition.

Theorem 3. For any natural \(n\), the asymptotic equality (9) for a bounded closed set \(A \subset R_n\) holds if and only if the \(n\)-dimensional Lebesgue measure of \(A\) is positive.

The last proposition complements the first of the estimates given in the above-mentioned table of A. N. Kolmogorov \((^1)\) (see \((^3)\), Theorem VIII). In the case when the Lebesgue measure of the boundary of the set \(A \subset R_n\) is zero (the set \(A\) has “volume”), excluding, for example, everywhere disconnected sets of positive measure, the asymptotic estimate of \(H_\varepsilon(A)\) and \(C_\varepsilon(A)\) as \(\varepsilon \to 0\) is known in a somewhat more precise form than (9) (see \((^3)\), § 4, Theorem IX).

Dnepropetrovsk Chemical-Technological
Institute

Received
27 I 1967

CITED LITERATURE

\(^1\) A. N. Kolmogorov, DAN, 108, No. 3, 385 (1956).
\(^2\) A. N. Kolmogorov, UMN, 10, issue 1, 192 (1955).
\(^3\) A. N. Kolmogorov, V. M. Tikhomirov, UMN, 14, issue 2 (86), 3 (1959).
\(^4\) A. G. Vitushkin, Estimate of the Complexity of the Tabulation Problem, Moscow, 1959.
\(^5\) Yu. A. Brudnyi, A. F. Timan, DAN, 126, No. 5, 927 (1959).
\(^6\) A. F. Timan, UMN, 19, issue 1 (115), 173 (1964).
\(^7\) A. F. Timan, Abstracts of the International Mathematical Congress, 5. Functional Analysis, Moscow, 1966, p. 76.
\(^8\) A. C. Vosburg, Proc. Am. Math. Soc., 17, No. 3, 665 (1966).
\(^9\) N. Bourbaki, Functions of a Real Variable, Moscow, 1965.
\(^10\) F. Hausdorff, Math. Ann., 79, 157 (1919).
\(^11\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.