Abstract
Full Text
MATHEMATICS
Yu. A. BRUDNYI, B. D. KOTLYAR
ON THE ORDER OF GROWTH OF THE $\varepsilon$-ENTROPY ON SOME COMPACT CLASSES OF FUNCTIONS
(Presented by Academician A. N. Kolmogorov on 17 VIII 1962)
- In the present note an estimate is given for the growth of the $\varepsilon$-entropy of certain functional compacta defined by their difference properties. The case of compacta defined by first differences of derivatives was studied by A. N. Kolmogorov ($^{3,5}$). The case of differences of second order was studied in ($^{2,6}$). In the present note, classes specified by difference properties of arbitrary order are studied systematically.
Let us agree on notation. Let $R^n$ be $n$-dimensional Euclidean space. The letters $x, y, z,\ldots$ will denote vectors from $R^n$; the letters $k, l, m,\ldots$—vectors with integral nonnegative components; $x_i, y_i, z_i,\ldots$ denote the $i$-th coordinates of the corresponding vectors. Indices referring to vectors will be placed as superscripts (for example, $x^m$). By $e^i$ we denote the vectors with components $\delta_j^i$, $j=1,2,\ldots,n$ ($\delta_j^i$ is the Kronecker symbol), $e=\sum_{i=1}^n e^i$.
Let $\omega_{k_i}^{(i)}(f;t)$ be the $k_i$-th modulus of smoothness of the function $f(x)$, defined in the domain $G$, taken with respect to the $i$-th variable, i.e.
\[ \omega_{k_i}^{(i)}(f;t) = \sup_{x\in G}\left|\Delta_{i,h}^{k_i}f(x)\right| = \sup_{\substack{x\in G\\ |h|\le t}} \left| \sum_{\nu=0}^{k_i}(-1)^{k_i-\nu} \binom{\nu}{k_i} f(x+\nu h e^i) \right|. \]
We denote by
\[ W=\left(\prod_{i=1}^{n}\!{}^{\otimes}\Delta_i^{k_i}(\varphi_i),\Pi,M\right)^{*} \]
the class of functions in the $n$-dimensional parallelepiped $\Pi=\prod_{i=1}^{n}[a_i,b_i]$ for which the following inequalities hold:
\[ \begin{aligned} &1)\quad |f(x)|\le M;\\ &2)\quad \omega_{k_i}^{(i)}(f;t)\le \varphi_i(t). \end{aligned} \tag{1} \]
We shall assume that the functions $\varphi_i(t)$ are given and do not decrease on $[0;+\infty)$, are continuous, and $\varphi_i(0)=0$.
The obtained class of functions is a compact** set in the space of continuous functions on $\Pi$ with the uniform metric. Denote by $H_\varepsilon(W)$ its $\varepsilon$-entropy (for the definition see ($^{3,9}$)).
* The notation of the class $W$ reflects the fact that it can be regarded as a topologized tensor product of the corresponding one-dimensional classes.
** By a compact set we shall everywhere mean a set that is completely bounded in $C$ (not necessarily closed).
- The \(\varepsilon\)-entropy of the compact set \(W\) admits a simple upper estimate.
Theorem 1. For each class \(W\) there exist positive constants \(C_1, C_2\), depending only on \(k\), such that
\[ H_{C_1\varepsilon}(W)\leq C_2\prod_{i=1}^{n}\frac{1}{\psi_i(\varepsilon)}+ O\left(\log_2\frac{1}{\varepsilon}\right), \]
where \(\psi_i(t)=\sup\{\alpha:\psi(t)\leq t\}\); if, however,
\[
\lim_{t\to0}\frac{\psi_i(t)}{t^k}=0,
\]
then we put \(\psi_i(t)\equiv1\).
For the proof we use the following lemma, which is a slight generalization of Whitney’s result (see \((^{10})\)).
Lemma. Let a bounded function \(f(x)\) be given on the \(n\)-dimensional parallelepiped
\[
\Pi=\prod_{i=1}^{n}[a_i,b_i],
\]
and let
\[
E_l(f;\Pi)=\inf_{P_l(x)}\sup_{x\in\Pi}|f(x)-P_l(x)|,
\]
where \(P_l(x)\) is a polynomial whose degree in the \(i\)-th variable is not higher than \(l_i\) \(\bigl(l=(l_i)_{i=1}^n\bigr)\). Then there exists a constant \(A_l\) (not depending on the function \(f\)) such that
\[
E_l(f;\Pi)\leq A_l\sum_{i=1}^{n}\omega^{(i)}_{l_i+1}
\left(f;\frac{b_i-a_i}{l_i+1}\right).
\]
- Passing to lower estimates of \(H_\varepsilon(W)\), let us note at once that, in order to obtain an effective result, it is necessary to impose certain restrictions on the functions \(\varphi_i(t)\). Indeed, as follows from the results of \((^8)\), if a function \(\varphi(t)\) satisfying the conditions
\[ \varphi(t)\geq Ct^k,\qquad \varphi(t)\leq C_1t^k,\qquad 0\leq t\leq b-a \]
(the second of the inequalities being satisfied at least in some neighborhood of zero), then the class \((\Delta^k(\varphi),[a,b],M)\), defined by this function, possesses \((k-1)\) derivatives which satisfy a Lipschitz condition with constant \(C_1\). Therefore, if we want to obtain a lower estimate coinciding with the upper estimate, we must take the majorants in some sense as the smallest possible; namely, if \(W=(\Delta^k(\varphi),[a,b],M)\), then by the smallest majorant we shall mean
\[ \overline{\varphi}(t)=\sup_{f\in W}\omega_k(f;t). \]
It is clear that \(\overline{\varphi}(t)\leq\varphi(t)\). But from the definition itself it is evident that \(\overline{\varphi}(t)\) must possess certain properties of a \(k\)-th modulus of smoothness. For \(k=1\) such characteristic properties are known (see, for example, \((^9)\)). Relying on this, it is not difficult to obtain the following estimate of \(H_\varepsilon(W)\) for the class \(W=(\Delta^1(\varphi_i),\Pi,M)\), where all \(\varphi_i\) are moduli of continuity:
\[ H_\varepsilon(W)\geq C_1\prod_{i=1}^{n}\frac{1}{\psi_i(C\varepsilon)} \]
(compare with Theorem 8 of \((^2)\)).
For moduli of smoothness of order \(k \geqslant 2\), we shall restrict ourselves to the most frequently occurring case of power majorants.
Let
\[ W=\left(\prod_{i=1}^{n}\otimes \Delta^{k_i}(t^{\beta_i}), \Pi, M\right)\quad (\beta_i \leqslant k_i), \]
where
\[ \Pi=\prod_{i=1}^{n}[a_i,b_i]. \]
Then
\[ H_{\varepsilon}(W)\geqslant C\varepsilon^{-\sum_{i=1}^{n}\frac{1}{\beta_i}} \]
(cf. the corresponding result of paper \((^5)\)).
Comparing the estimate obtained with the result of Theorem 1, we obtain the following assertion.
Theorem 2. The exact order of growth of the \(\varepsilon\)-entropy of the compact set
\[ W=\left(\prod_{i=1}^{n}\otimes \Delta^{k_i}(t^{\beta_i}),\Pi,M\right)\quad (\beta_i \leqslant k_i) \]
is equal to \(\varepsilon^{-\sum_{i=1}^{n}\frac{1}{\beta_i}}\), i.e.,
\[ H_{\varepsilon}(W)\asymp \varepsilon^{-\sum_{i=1}^{n}\frac{1}{\beta_i}}. \tag{2} \]
Using the results of papers \((^8,^1)\), one can give a more convenient description of the classes
\[ W=\left(\prod_{i=1}^{n}\otimes \Delta^{k_i}(t^{\beta_i}),\Pi,M\right)\quad(\beta_i \leqslant k_i). \]
Namely, it follows from these papers that if \(\omega_k(f;t)\leqslant t^\beta\) \((\beta\leqslant k)\), then:
a) for \(\beta=k\), \(f(x)\) has a derivative of order \(s=\beta-1\) satisfying the condition
\[ \left|\Delta_h^1 f^{(s)}(t)\right|\leqslant h; \]
b) for fractional \(\beta\), \(f(x)\) has \(s=[\beta]\) derivatives, and
\[ \left|\Delta_h^1 f^{(s)}(t)\right|\leqslant C(\beta)h^{\beta-s}; \]
c) for integral \(\beta\ne k\), \(f(x)\) has a derivative of order \(s=\beta-1\), and
\[ \left|\Delta_h^2 f^{(s)}(t)\right|\leqslant C(\beta)h. \]
Hence it follows that the class \(W\) consists of functions having, in each variable, the corresponding number of derivatives which, depending on the relations between \(\beta_i\) and \(k_i\), satisfy one of the inequalities in items a)—c).
In particular, for \(n=1\), \(\beta=k-\alpha\) \((0\leqslant\alpha\leqslant1)\), it follows from (2) that the \(\varepsilon\)-entropy of the corresponding compact set (which consists of uniformly bounded functions differentiable \(k-1\) times, with the \((k-1)\)-st derivative satisfying the Hölder condition of order \(1-\alpha\)) has order \(\varepsilon^{-1/\beta}\). This result was obtained earlier by A. N. Kolmogorov \((^3)\).
If \(n=1\), \(\beta=k-1\), then the \(\varepsilon\)-entropy of the corresponding compact set, consisting of functions differentiable \(k-2\) times whose \((k-2)\)-nd derivative is quasi-smooth, has order \(\varepsilon^{-1/\beta}\). For \(k=2\), this result was obtained earlier by B. D. Kotlyar (see \((^6)\)).
Remark 1. Consider the class
\[ W=\left(\prod_{i=1}^{n}\otimes \Delta^{k_i}(t_i^{\beta_i}),\Pi,M\right), \]
where \(k_i=s\), \(\beta=s-\alpha\) \((0\leqslant\alpha\leqslant1)\). Then the corresponding estimate has the form
\[ H_{\varepsilon}(W)\asymp \varepsilon^{-\frac{n}{s-\alpha}}. \]
It is interesting to compare this result with the result obtained by A. N. Kolmogorov. In (³) it is proved that
\[ H_{\varepsilon}(W_1') \asymp \varepsilon^{-\frac{n}{s-\alpha}}, \]
where \(W_1'\) is the class of functions defined on \(\Pi\) and possessing mixed derivatives up to order \((s-1)\) inclusive, satisfying a Hölder condition of order \(1-\alpha\) \((0<\alpha<1)\). Since in the definition of \(W\) the existence of derivatives of order \((s-1)\) is required only with respect to each variable separately (they satisfy a Hölder condition of order \(1-\alpha\)), the compact set \(W\), generally speaking, is broader than \(W_1'\). However, if these classes consist of periodic functions, then, by virtue of Montel’s result (see (⁷)), the indicated classes coincide (see also (⁹)). In the nonperiodic case, however, it has been proved only that the corresponding mixed derivatives exist in any closed domain \(G\) lying strictly inside the parallelepiped. But, apparently, in reality the indicated classes coincide also in the nonperiodic case.
Remark 2. Estimate (2) makes it possible to strengthen A. G. Vitushkin’s result on the impossibility of representing a function of several variables by functions of a smaller number of variables. Namely, repeating A. N. Kolmogorov’s reasoning (see (⁴)), one can prove this theorem for the case when the functions composing the superposition have different differentiability properties with respect to different variables.
Received
17 VIII 1962
REFERENCES
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