MATHEMATICS

V. M. TIKHOMIROV

ON \(n\)-DIMENSIONAL WIDTHS OF CERTAIN FUNCTIONAL CLASSES

(Presented by Academician A. N. Kolmogorov on 14 IX 1959)

1. Let \(F\) be a set lying in a Banach space \(R\), consisting of elements \(\{x,y,\ldots,f,\ldots\}\); let \(L_n\) be some linear \(n\)-dimensional subspace. We introduce the quantities

\[ \rho(x,L_n)=\inf_{y\in L_n}\|x-y\|,\qquad \delta(F,L_n)=\sup_{x\in F}\rho(x,L_n). \]

The latter quantity will be called the deviation of the set \(F\) from the subspace \(L_n\). The lower bound of \(\delta(F,L_n)\) over all subspaces \(L_n\) of dimension \(n\) will be called the \(n\)-dimensional width of the set \(F\) and will be denoted by \(d_n(F)\). \(n\)-Dimensional widths for functional classes were introduced by A. N. Kolmogorov in the paper \((^1)\), where they were computed for certain classes of functions in the metric \(L^2\).

We prove the following theorem on \(n\)-dimensional widths*.

Theorem. Let \(U\) be the unit sphere in the space \(R\); let \(F_{n+1}\) and \(U_{n+1}\) be sections of the sets \(F\) and \(U\) by some \((n+1)\)-dimensional subspace \(L_{n+1}\). Then, if \(\alpha U_{n+1}\subseteq F_{n+1}\), then \(d_n(F)\geq \alpha\).

Using this theorem, we compute two \(n\)-dimensional widths in the metric \(C\) of the following classes, known in approximation theory, of \(2\pi\)-periodic functions with mean value zero:

a) the class \(F_r\) of real functions \(f(x)\) for which
\[ \operatorname{vrai\,max}|f^{(r)}(x)|\leq 1; \]

b) the class \(A_h\) of functions \(f(z)=f(x+iy)\), real on the real axis, regular in the strip \(-h<y<+h\) and satisfying there the inequality \(|\operatorname{Re} f(z)|<1\);

c) the class \(\Gamma_\rho,\ \rho<1\), of functions \(f(\theta)=u(\rho,\theta)\), \(0\leq\theta\leq 2\pi\), where the functions \(u(r,\theta)\), \(0\leq r\leq 1,\ 0<\theta\leq 2\pi\), are harmonic in the disk of radius one and satisfy in it the inequality \(|u(r,\theta)|<1\),

and also

d) the class \(B_k\) of functions \(f(z)\), analytic in the disk \(|z|<1\), for which \(|f^{(k)}(z)|<1,\ |z|<1\).

We shall show that:

\[ d_{2n}(F_r)=\delta(F_r,T_{2n}) = \frac{4}{\pi(n+1)^r} \sum_{m=0}^{\infty} \frac{(-1)^m(r+1)}{(2m+1)^r+1}; \tag{I} \]

\[ d_{2n}(A_h)=\delta(A_h,T_{2n}) = \frac{4}{\pi} \sum_{m=0}^{\infty} (-1)^m \frac{1}{(2m+1)} \frac{1}{\operatorname{ch}(2m+1)(n+1)h}; \tag{II} \]

\[ d_{2n}(\Gamma_\rho)=\delta(\Gamma_\rho,T_{2n}) = \frac{4}{\pi}\operatorname{arc\,tg}\rho^{\,n+1}. \tag{III} \]

\[ \text{* The expression } x+\gamma U,\ \text{where } \gamma>0,\ U \text{ is a set, denotes here the set of points representable in the form } x+\gamma z,\ z\in U. \]

The values of the deviations for the listed classes from the space \(T_{2n}\) of polynomials

\[ \sum_{k=1}^{n}(a_k\cos kx+b_k\sin kx) \]

were computed long ago in the works of J. Favard \((^{2})\) and N. I. Akhiezer and M. G. Krein \((^{3})\) for the class \(F_r\), of N. I. Akhiezer \((^{4})\) for the class \(A_h\), and of M. G. Krein \((^{5})\) for the class \(\Gamma_\rho\).

Relations (I)—(III) show that the subspace \(T_{2n}\) is “closest,” in the metric \(C\), to the sets \(F_r\), \(A_h\), and \(\Gamma_\rho\) among all possible linear subspaces of dimension \(2n\).* For the class \(B_k\) the following relation holds:

\[ d_n(B_k)=\delta(B_k,P_n)=\frac{1}{n(n-1)\ldots(n-k+1)}. \]

The value of the deviation of the class \(B_k\) from the space \(P_n\) of algebraic polynomials was computed in a recent work of K. I. Babenko \((^{6})\).

In § 4 we give formulas for \(n\)-dimensional widths and their estimates for the classes \(F_q^{E_s}\) of functions of \(s\) variables of smoothness \(q\) and for the class of functionals \(F_{\beta}^{\widetilde F_1^{[a,b]}}\), for which there is no developed theory of approximation by any fixed subspace.

1. We proceed to the proof of the theorem. The \(n\)-dimensional space \(L_n\) appearing below is regarded as realized as a coordinate \(n\)-dimensional space with points \(y=(y_1,\ldots,y_n)\). The zero vector of \(L_n\) will be denoted by \(\theta\). In \(L_n\) Lebesgue measure is assumed to be specified.

Lemma. Let \(U_N\) be a convex centrally symmetric body in the affine \(N\)-dimensional space \(R_N\) with center at zero; let \(U_{n+1}\) be the section of \(U_N\) by some fixed \((n+1)\)-dimensional subspace \(L_{n+1}^{*}\subset R_N\), \(n+1\le N\); let \(S_n\) be the boundary of \(U_{n+1}\). Then for every \(n\)-dimensional subspace \(L_n\) there exists a point \(x_0=x_0(L_n)\in S_n\) such that the hyperplane \(x_0+L_n\) is a supporting hyperplane of \(U_n\), i.e., for every \(\varepsilon>0\)

\[ (1+\varepsilon)x_0+L_n\cap U_N=0. \tag{A} \]

The convex centrally symmetric body \(U_N\) specified in the lemma defines in \(R_N\) a Banach metric in which it itself is the sphere. Relation (A) means that in this metric \(\rho(x_0,L_n)=1\).

Thus our lemma can be reformulated as follows: the \(n\)-dimensional width of an \(n\)-dimensional sphere does not decrease when it is embedded in a subspace of a larger number of dimensions.

Proof of the lemma. The Banach metric defined by the body \(U_N\) determines a certain topology in \(R_N\). Consider the following nonnegative function of two variables \(x\in S_n\) and \(y\in L_n\):

\[ \varphi(x,y)= \begin{cases} \max \gamma \text{ over all } \gamma \text{ such that } x+y+\gamma \overline U_N\in \overline U_N,\\ \qquad\qquad\qquad\qquad\qquad\qquad\text{if } x+y\in \overline U_N,\\ 0,\qquad\qquad\qquad\qquad\qquad\quad\;\;\text{if } x+y\in R_N\setminus \overline U_N. \end{cases} \tag{1} \]

The function \(\varphi(x,y)\) is continuous. Indeed, if \(\varphi(x_0,y_0)=\alpha>0\), then for any \(\delta>0\) and \(x+y\in x_0+y_0+\delta U_N\) we obtain that \(\alpha-\delta\le \varphi(x,y)\le \alpha+\delta\); whereas if \(\varphi(x_0,y_0)=0\), then for any \(\delta\) and \(x+y\in x_0+y_0+\delta U_N\) we obtain \(0\le \varphi(x,y)\le 2\delta\).

Now put

\[ \psi(x)=\max_{y\in L_n}\varphi(x,y). \]

It is clear that \(\psi(x)\) is continuous on \(S_n\). The assertion of the lemma is equivalent to the existence of a point \(x_0\in S_n\) such that \(\psi(x_0)=0\). Suppose the contrary. Let \(\psi(x)>0\) for every point \(x\in S_n\), and consequently, by the continuity of \(\psi(x)\) and the compact-

\[ \text{* We note in this connection that the space } T_{2n}, \text{ generally speaking, is not the only subspace satisfying the extremal property (I).} \]

of \(S_n\), \(\psi(x) \geqslant \alpha > 0\). For each \(x\) consider the set \(G(x) \subset L_n\), evidently nonempty, such that if \(y \in G(x)\), then

\[ \varphi(x,y)\geqslant \tfrac12\alpha . \tag{2} \]

From (2) and the continuity of \(\varphi(x,y)\) follows the closedness of the set \(G(x)\). The set \(G(x)\) is convex. Indeed, if \(y_1,y_2\in G(x)\), then this means that
\[ x+y_1+\tfrac12\alpha \overline U_N \subset \overline U_N,\qquad x+y_2+\tfrac12\alpha \overline U_N \subset \overline U_N. \]
But then also
\[ x+\gamma_1y_1+\gamma_2y_2+\tfrac12\alpha \overline U_N\in \overline U_N,\qquad \gamma_1+\gamma_2=1,\quad \gamma_1\geqslant0,\quad \gamma_2\geqslant0, \]
in view of the convexity of the set \(U_N\). Moreover, the set \(G(x)\) contains the zero vector \(\theta\) for no \(x\), since, by (1), \(\varphi(x,\theta)=0\), and, in consequence of the equality \(\varphi(\bar x,-y)=\varphi(x,y)\), where \(\bar x\) denotes the point symmetric to \(x\) with respect to the center (recall that \(U_N\), and hence also \(S_n\), are centrally symmetric), we obtain that the sets \(G(x)\), as functions of \(x\), are cosymmetric: \(G(\bar x)=-G(x)\). Finally, from the continuity of the function \(\varphi(x,y)\) it is easy to derive the continuous dependence of the set \(G(x)\) on \(x\). In view of the convexity of \(G(x)\), all these sets are measurable.

Let \(\eta(x)\) denote the center of gravity of the set \(G(x)\). From what has been said above it follows that:

a) \(\eta(x)\) is a vector of \(n\) dimensions, continuously depending on \(x\);

b) \(\eta(x)\neq \theta\) at no point \(x\in S_n\);

c) \(\eta(\bar x)=-\eta(x)\).

But, according to Borsuk’s theorem \((^7)\), on the \(n\)-dimensional sphere there cannot exist an \(n\)-dimensional vector field satisfying properties a)—c).*

We have arrived at a contradiction. The lemma is proved. The theorem follows simply from the lemma. Let \(L_n\) be some linear subspace. Consider the minimal subspace \(R_N\) such that it contains the subspace \(L_{n+1}^*\) of the theorem and the given subspace \(L_n\). Let \(U_N\) denote the intersection of the unit sphere \(U\) with the subspace \(R_N\).

According to the lemma, there will be found a point \(x_0\in \alpha \overline U_{n+1}\subset F_{n+1}\subset F\) such that, for any \(\varepsilon>0\),
\[ (1+\varepsilon)x_0+L_n\cap \alpha \overline U_N=\varnothing \]
and, consequently,

\[ (1+\varepsilon)x_0+L_n\cap \alpha U=0. \tag{3} \]

It is easy to understand that relation (3) means that
\[ \rho(x_0,L_n)\geqslant \frac{\alpha}{1+\varepsilon}. \]
In view of the arbitrariness of \(\varepsilon\) and \(L_n\), we obtain what was required in the theorem.

1. We shall now derive relation (1). Let \(f_0(x)\) denote the function from \(F_r\) for which
   \[ f_0^{(r)}(x)=\operatorname{sgn}\sin nx. \]
   The function \(f_0(x)\) has the following properties:

a)
\[ \max_{x\in[0,2\pi]} |f_0(x)| = \frac{4}{\pi n^r} \sum_{m=0}^{\infty} \frac{(-1)^{m(r+1)}}{(2m+1)^{r+1}}; \]

b) \(f_0(x)\) assumes its maximum value with alternating signs at \(2n\) consecutive points of the interval \([0,2\pi)\):
\[ x_k=\frac{k\pi}{n}+\frac{\pi\varepsilon}{2} \]
\((k=0,1,\ldots,2n-1)\), \(\varepsilon=1\), if \(r\) is even, and \(\varepsilon=0\), if \(r\) is odd. Denote by \(\Delta_k\) the interval
\[ k\pi\leqslant x\leqslant (k+1)\pi,\qquad k=0,1,\ldots,2n-1. \]
Consider \(2n\) piecewise-constant functions:
\[ \varphi_k(x)=1,\quad \text{if } x\in \Delta_k;\qquad \varphi_k(x)=0,\quad \text{if } x\notin \Delta_k. \]
Let \(L_{2n-1}^*\) denote the \((2n-1)\)-dimensional space of functions \(f(x)\) for which
\[ f^{(r)}(x)=\sum_{k=0}^{2n-1} c_k\varphi_k(x),\qquad \sum_{k=0}^{2n-1} c_k=0. \]

* We note that Borsuk’s theorem can be easily derived from the well-known theorem of Lusternik–Schnirelmann on the covering of spheres, as is done in \((^8)\), p. 107.

we prove that for every function \(f(x)\in L_{2n-1}^{*}\) for which \(\left|f^{(r)}(x)\right|\leqslant 1,\ 0\leqslant x\leqslant 2\pi\), and on some \(\Delta_k\), \(\left|f^{(r)}(x)\right|=1\), the inequality

\[ \max_{x\in[0,2\pi]} |f(x)| \geqslant \max_{x\in[0,2\pi]} |f_0(x)|. \tag{4} \]

is satisfied.

The relations (I) are derived automatically from (4) and the theorem. Suppose, without loss of generality, that \(f^{(r)}(x)=+1\) for \(x\in\Delta_0\). We split the function \(f(x)\) into two: \(f(x)=f_0(x)+f_1(x)\).

Assume that for some function \(f(x)\) the inequality opposite to (4) holds. Then the function \(f_1(x)\) will have \(2n\) sign changes at the points \(x_k\). Applying Lagrange’s theorem \(r-1\) times, we obtain that the function \(f^{(r-1)}(x)\) also has \(2n\) sign changes, which is impossible, since it is piecewise linear on the \(2n-1\) intervals \(\Delta_k,\ k=1,2,\ldots,2n-1\), and constant on \(\Delta_0\).

The equalities (II) and (III) can be obtained on the basis of analogous arguments.

1. A real functional \(F(f)\) on a set \(A\) from a Banach space \(R\) will be called satisfying the Hölder condition with exponent \(\beta,\ 0<\beta\leqslant 1\), if
   \[ |F(f)-F(f')|\leqslant \|f-f'\|^{\beta}. \]
   By \(F_\beta^A\) we denote the set of all such functionals.

In particular, \(F_\beta^{[a,b]}\) denotes the set of functions on the interval \([a,b]\) satisfying the usual Hölder condition

\[ |f(x)-f(x')|\leqslant |x-x'|^\beta . \]

\(F_\beta^{E_s}\) is the set of functionals satisfying the Hölder condition with exponent \(\beta\) on the \(s\)-dimensional cube \(E_s:\ |x_k|\leqslant 1/2,\ k=1,2,\ldots,s\), with metric

\[ \|x-x'\|=\max_{1\leqslant k\leqslant s}|x_k-x'_k|. \]

\(\widetilde{F}_{\beta}^{[a,b]}\) is the set of functionals satisfying the Hölder condition with exponent \(\beta\) on the space \(\widetilde{F}_1^{[a,b]}\) of functions \(f(x)\) satisfying on \([a,b]\) the Lipschitz condition \((\beta=1)\) with \(f(a)=0\).

Finally, let \(F_q^{E_s}\) denote the set of functions \(f(x_1,\ldots,x_s)\) of \(s\) variables, of smoothness \(q\) in the sense of (9) (p. 32).

The following relations hold \(\bigl(f(n)\asymp g(n)\) for positive functions \(f\) and \(g\) of \(n\) means that as \(n\to\infty\), \(f=O(g)\) and \(g=O(f)\bigr)\):

\[ \begin{aligned} \text{a)}\quad & d_n\!\left(F_\beta^{[a,b]}\right)=\left(\frac{b-a}{2n}\right)^\beta; & \qquad \text{b)}\quad & d_n\!\left(F_\beta^{E_s}\right)=\left(\frac{1}{2[n]^{1/s}}\right)^\beta;\\[6pt] \text{c)}\quad & d_n\!\left(F_q^{E_s}\right)\asymp \frac{1}{n^{q/s}}; & \qquad \text{d)}\quad & d_n\!\left(\widetilde{F}_{\beta}^{[a,b]}\right)=\left\{\frac{b-a}{2\left([ \log_2 n]+1\right)}\right\}^{\beta}. \end{aligned} \]

Relation c) is a generalization of the relation \(d_n(F_q^{E_1})\asymp 1/n^q\), obtained by S. B. Stechkin (10).

In conclusion I express my deep gratitude to A. N. Kolmogorov for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
10 IX 1959

CITED LITERATURE

1. A. Kolmogoroff, Ann. of Math., 37, No. 1, 107 (1936).
2. J. Favard, Bull. Sci. Math., 61, 243 (1937).
3. N. I. Akhiezer, M. G. Krein, DAN, 15, No. 2, 107 (1937).
4. N. I. Akhiezer, DAN, 18, No. 4–5, 241 (1938).
5. M. G. Krein, DAN, 18, No. 4–5, 245 (1938).
6. K. I. Babenko, Izv. AN SSSR, ser. matem., 22, No. 5 (1958).
7. K. Borsuk, Fund. Math., 20, 177 (1933).
8. M. A. Krasnosel’skii, Topological Methods..., Moscow, 1956.
9. A. N. Kolmogorov, V. M. Tikhomirov, Uspekhi Mat. Nauk, 14, No. 2 (86), 3 (1959).
10. S. B. Stechkin, Uspekhi Mat. Nauk, 9, No. 1 (59), 133 (1954).