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MATHEMATICS
S. Ya. AL′PER
ON THE ε-ENTROPY OF CERTAIN CLASSES OF FUNCTIONS
(Presented by Academician A. N. Kolmogorov, 26 II 1960)
Let \(T\) be a compact set in a metric space \(X\), and let \(H_\varepsilon(T)\) be the \(\varepsilon\)-entropy of the set \(T\), \(H_\varepsilon(T)=\lg N_\varepsilon(T)\), where \(N_\varepsilon(T)\) is the smallest number of elements of \(X\) forming an \(\varepsilon\)-net for \(T\), and the logarithm is taken to base 2. We shall use the signs \(\asymp\) and \(\sim\) for weak and strong equivalence, respectively, and also the signs \(\ll\) and \(\gg\), adopted in \((^1)\).
Let \(G\) be a finite simply connected domain of the \(z\)-plane with rectifiable boundary \(\Gamma\), whose parametric equation is
\(z=z(s)=x(s)+iy(s)\), \(0\leq s\leq L\), where \(s\) is the arc length on \(\Gamma\), measured from some point. The belonging of the curve \(\Gamma\) to the class \(C_\mu^m\) means that \(x(s)\) and \(y(s)\) have derivatives up to order \(m\) inclusive for \(0\leq s\leq L\), and the derivatives of order \(m\) satisfy a Lipschitz condition in \(s\) with exponent \(\mu\), \(0<\mu\leq 1\).
\(1^\circ\). Consider the question of the \(\varepsilon\)-entropy of a class of functions analytic in the domain \(G\) and continuous in the closed domain \(\overline{G}\), with additional conditions on the character of continuity in \(\overline{G}\). We define the norm of a function \(f(z)\) by the equality
\[
\|f\|=\max_{z\in \overline{G}} |f(z)|.
\]
Lemma 1. If the boundary of the domain \(G\) is a rectifiable Jordan curve and \(T_q\) is the class of functions analytic in the domain \(G\), continuous together with their derivatives up to order \(p\) inclusive in \(\overline{G}\), and satisfying the conditions
\[
|f^{(k)}(z)|\leq C_k \quad (k=0,1,2,\ldots,p), \qquad z\in \overline{G};
\tag{1}
\]
\[
|f^{(p)}(z_1)-f^{(p)}(z_2)|\leq C|z_1-z_2|^\alpha, \qquad 0<\alpha\leq 1,
\tag{2}
\]
for any points \(z_1\) and \(z_2\in\Gamma\), where \(C_k\) and \(C\) are given constants \(>0\), then
\[
H_\varepsilon(T_q)\ll \left(\frac{1}{\varepsilon}\right)^{1/q}, \qquad q=p+\alpha.
\tag{3}
\]
Explaining the course of the proof, define the number
\(\Delta=(\varepsilon/2C)^{1/q}\) and consider the values of the arc
\(s_0=0,\ s_r=s_0+r\Delta,\ r=0,1,2,\ldots,m\), where \(m\) is such that
\(s_m\leq L,\ s_{m+1}>L\); the corresponding points on \(\Gamma\) will be
\(z_r=z(s_r)\). Put
\[
\beta_r^{(k)}(f)=\left[\frac{f^{(k)}(z_r)}{\varepsilon_k}\right],
\qquad k=0,1,2,\ldots,p;\quad r=0,1,2,\ldots,m,
\]
where
\[
\varepsilon_k=\frac{\varepsilon}{2e\Delta^k}, \qquad [a+bi]=[a]+[b]\,i,
\]
and associate with the function \(f(z)\in T_q\) the matrix
\[
\beta=\|\beta_r^{(k)}\|.
\]
Further on we use the method applied by A. N. Kolmogorov in paper \((^{1})\) (§ 5), but the estimate of the remainder term of Taylor’s formula is made by representing it as an integral over the contour \(\Gamma\).
Lemma 2. If the boundary \(\Gamma \in C_\mu^1,\ 0<\mu\leqslant 1\) (a Lyapunov curve), then the following lower estimate holds
\[ H_\varepsilon(T_q)\succcurlyeq \left(\frac{1}{\varepsilon}\right)^{1/q}, \qquad 0<\alpha<1. \tag{4} \]
The proof can be obtained with the aid of Theorem 1 from paper \((^{2})\), using the direct theorem on best approximation in a complex domain \((^{3})\) and the inverse theorem of S. N. Mergelyan \((^{4})\) for a domain with a boundary of the type under consideration.
Theorem 1. If \(\Gamma \in C_\mu^1,\ 0<\alpha<1\), then
\[ H_\varepsilon(T_q)\asymp \left(\frac{1}{\varepsilon}\right)^{1/q}. \tag{5} \]
In the case \(\alpha=1\) the relation
\[ \lg H_\varepsilon(T_q)\sim \frac{1}{q}\lg\frac{1}{\varepsilon} \tag{6} \]
is valid.
Let us note that the theorem remains valid under the condition on \(\Gamma\) that is somewhat less restrictive:
\[ \int_0^c \frac{j(h)}{h}|\lg h|\,dh<\infty, \]
where \(j(h)\) is the modulus of continuity of the function \(z'(s)\).
The following two assertions are also valid.
Theorem 2. If the boundary of the domain \(G\) is an arbitrary smooth curve with continuously rotating tangent and \(0<\alpha\leqslant 1\), then
\[ \lg H_\varepsilon(T_q)\sim \frac{1}{q}\lg\frac{1}{\varepsilon}. \]
Theorem 3. If the boundary \(\Gamma\) of the domain \(G\) is an arbitrary rectifiable Jordan curve and \(0<\alpha\leqslant 1\), then\(^*\)
\[ \lg H_\varepsilon(T_q)\asymp \lg\frac{1}{\varepsilon}. \]
\(2^\circ\). Let us consider the question of the \(\varepsilon\)-entropy for the class of generalized analytic functions \((^{5})\). Suppose the equation
\[ \frac{dw}{d\bar z}=A(z)w+B(z)\bar w=0, \tag{7} \]
is given, where \(w(z)=u(x,y)+iv(x,y),\ z=x+iy\), and
\[ \frac{\partial w}{\partial \bar z} = \frac{1}{2}\left(\frac{\partial w}{\partial x} +i\frac{\partial w}{\partial y}\right). \]
Such an equation represents the writing, in complex form, of the elliptic system of equations
\[ \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=au+bv,\qquad \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=cu+dv, \tag{8} \]
whose coefficients depend on \(x\) and \(y\). We shall say that a function \(\Phi(z)\in C_\mu^m(\bar G)\) if \(\Phi(z)\) and all its partial derivatives with respect to \(x\) and \(y\) up to order \(m\) inclusive are continuous in \(\bar G\), and all partial derivatives of order \(m\) satisfy in the closed domain \(\bar G\) a Lipschitz condition of order—
\(^*\) The upper estimate for \(H_\varepsilon(T_q)\) follows from Lemma 1; the lower estimate can be obtained in the form
\[ H_\varepsilon(T_q)\geq A\left(\frac{1}{\varepsilon}\right)^{\frac{1}{2p+2}}, \]
where \(A\) is a constant.
with respect to \(z\) with exponent \(\mu,\ 0<\mu\leqslant 1\). The norm of the function \(w(z)\) is defined by the equality \(\|w\|=\max\limits_{z\in \overline G}|w(z)|\).
Theorem 4. If \(\Gamma\in C_\alpha^p,\ 0<\alpha<1,\ A\) and \(B\in C_\alpha^p(\overline G)\), then for the class \(W_q\) of solutions \(w(z)\) of equation (7) satisfying the conditions
\[ |w(z)|\leqslant C_0,\qquad \left|\frac{\partial^k w}{\partial x^s\partial y^{k-s}}\right|\leqslant C_k,\qquad s=0,1,\ldots,k,\quad k=0,1,\ldots,p, \]
\(z\in \overline G\), and the requirement that the Lipschitz constant with exponent \(\alpha\) for each partial derivative of order \(p\) of \(w\) not exceed \(C\) (\(C_k\) and \(C>0\)), the relation
\[ H_\varepsilon(W_q)\asymp \left(\frac{1}{\varepsilon}\right)^{1/q},\qquad q=p+\alpha, \tag{9} \]
holds. For \(\alpha=1\) we have
\[ \lg H_\varepsilon(W_q)\sim \frac{1}{q}\lg\frac{1}{\varepsilon}. \tag{10} \]
Let us note that the functions of the class \(W_q\) satisfy equation (7) in the domain \(G\) in the classical sense.
The upper estimate is based on the maximum-modulus principle for generalized analytic functions: \(|f(z)|\leqslant M\max\limits_{z\in \Gamma}|f(z)|,\ z\in G\) (the constant \(M\) depends only on the functions \(A\) and \(B\))—and on the method of A. N. Kolmogorov \((^1)\). The lower estimate is based on the existence of a solution of the boundary-value problem \(\operatorname{Re} w(z)=\gamma(z)\) for equation (7) with an arbitrary sufficiently smooth function \(\gamma(z)\) and on known theorems \((^5)\) on the dependence between the smoothness character of \(\gamma(z)\) and the solution \(w(z)\).
\(3^\circ\). Let us also consider the question of the \(\varepsilon\)-entropy of the class \(V_\mu\) of entire analytic functions \(f(z)\), whose growth satisfies the condition
\[ |f(z)|\leqslant e^{\mu(r)} \]
on the circle \(|z|=r\) for all \(r>r_0\), where \(r_0\) depends only on \(\mu(r)\). Suppose that \(\mu(r)\uparrow\infty\) and \(r\mu'(r)\uparrow\infty\) as \(r\uparrow\infty\), and that the condition \(r\mu''(r)[\mu'(r)]^{-1}\geqslant d>-1\) is fulfilled for \(r>r_1\); let \(\|f\|=\max\limits_{|z|=R}|f(z)|\). Denote by \(r=h(n)\) the solution of the equation \(z\mu'(r)=n\), and by \(q(t)\) the solution of the equation \(n\lg h(n)=t\). In both cases it suffices to find an asymptotic solution.
Theorem 5. The following relation holds:
\[ H_\varepsilon(V_\mu)\sim q\left(\lg\frac{1}{\varepsilon}\right)\lg\frac{1}{\varepsilon}. \tag{11} \]
If, in particular, we set \(\mu(r)=\sigma r^\rho\), then we obtain the result of A. G. Vitushkin \((^1)\)
\[ H_\varepsilon(V_\mu)\sim \rho\, \frac{\left(\lg\frac{1}{\varepsilon}\right)^2}{\lg\lg\frac{1}{\varepsilon}}. \]
If we set \(\mu(r)=e^{\sigma r^\rho}\), then we shall have
\[ H_\varepsilon(V_\mu)\sim \rho\, \frac{\left(\lg\frac{1}{\varepsilon}\right)^2}{\lg\lg\lg\frac{1}{\varepsilon}}. \]
Rostov State University
Received
25 II 1960
References
- A. N. Kolmogorov, V. M. Tikhomirov, Uspekhi Mat. Nauk, 14, 2(86) (1959).
- Yu. A. Brudnyi, A. F. Timan, Dokl. Akad. Nauk SSSR, 126, No. 5, 931 (1959).
- S. Ya. Al’per, Izv. Akad. Nauk SSSR, Ser. Mat., 19, 423 (1955).
- S. N. Mergelyan, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 37 (1951).
- I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.