MATHEMATICS

A. F. TIMAN

ON A GEOMETRIC PROBLEM IN APPROXIMATION THEORY

(Presented by Academician A. N. Kolmogorov on 27 IV 1961)

1. General formulation of the question. Consider a metric space \(R\), an arbitrary set \(G\) in it, some bounded set \(F\), and the quantity

\[ \mathscr{E}_{G}(F)=\sup_{f\in F}\inf_{g\in G}\rho(f,g), \tag{1} \]

which is called the deviation of \(F\) from \(G\).

In the constructive theory of functions, various versions of problems connected with determining the upper bound \(\mathscr{E}_{G}(F)\) are well known. In these problems \(R\) is usually a linear normed space. At present a number of exact or asymptotically exact results are known for the case when \(G\) is a finite-dimensional subspace of \(R\) (the case of approximation by “polynomials” of a given order). Some estimates of this kind are also available in other cases. We shall cite one well-known result of M. G. Krein, corresponding to the space of functions \(f(t)\) that are bounded and uniformly continuous on the entire real axis, where \(\rho(f,g)=\sup_{-\infty<t<\infty}|f(t)-g(t)|\).

If \(F\) consists of all \(r\)-times differentiable functions for which \(\sup_{-\infty<t<\infty}|f^{(r)}(t)|\le M\), and \(G\) is the set of all entire functions of degree \(\le \sigma\), then (see \((^{1,7})\))

\[ \mathscr{E}_{G}(F)=\frac{4M}{\pi\sigma^{r}}\sum_{k=0}^{\infty}\frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}}. \tag{2} \]

In the present note the quantity (1) is considered for some other sets*.

2. An elementary example for the plane. Let \(R_{2}\) be the two-dimensional real plane, where the distance between the points \(P(x',y')\) and \(Q(x'',y'')\) is defined by the formula \(\rho(P,Q)=\max\{|x'-x''|,\ |y'-y''|\}\); let \(F_b\) be the set of all points whose coordinates satisfy the condition \(|x-y|\le b\), where \(c<\infty\) is some positive constant. If \(F\) is the intersection of \(F_b\) with the square \(|x|\le c,\ |y|\le c\), and \(G\) is the intersection of this square with \(F_a\), then it is obvious that for \(0\le a\le b\le c\) the equality

\[ \mathscr{E}^{R_{2}}_{G}(F)=\frac{1}{2}(b-a). \tag{3} \]

holds.

3. An analogous example for \(m\)-dimensional space. Consider the \(m\)-dimensional real space \(R_m\), where the distance between the points \(P(x'_1,\ldots,x'_m)\) and \(Q(x''_1,\ldots,x''_m)\) is defined by the formula \(\rho(P,Q)=\max_{1\le k\le m}|x'_k-x''_k|\). Let \(B\) and \(A\) be, respectively—

* The contents of this note were presented in a report by the author at the seminar on the theory of functions at Dnepropetrovsk University on March 28, 1961.

respectively, two symmetric square matrices of \(m^2\) real numbers \(\{b_{\nu k}\}\) and \(\{a_{\nu k}\}\) such that \(b_{\nu\nu}=a_{\nu\nu}=0\), \(a_{\nu k}>0\) \((\nu\ne k)\), \(a_{\nu k}\le a_{\nu s}+a_{ks}\), \(b_{\nu k}>0\) \((\nu\ne k)\), \(b_{\nu k}\le b_{\nu s}+b_{ks}\), and \(c<\infty\) is a certain constant satisfying the condition \(\max_{1\le \nu,k\le m} a_{\nu k}\le c\), \(\max_{1\le \nu,k\le m} b_{\nu k}\le c\). Denote by \(F_{b_{\nu k}}\) the set of all points \(R_m\) for which \(|x_\nu-x_k|\le b_{\nu k}\), and by \(F_B\) the intersection of all \(F_{b_{\nu k}}\) \((\nu,k=1,\ldots,m)\). If \(F\) is the intersection of \(F_B\) with the \(m\)-dimensional “sphere” of radius \(c\) whose center is at zero, and \(G\) is the intersection of this sphere with \(F_A\), then for the polyhedra \(F\) and \(G\) the equality holds

\[ \mathscr{E}^{R_m}_{G}(F)=\frac{1}{2}\max_{1\le \nu,k\le m}(b_{\nu k}-a_{\nu k}). \tag{4} \]

1. Best uniform approximation of functions in a prescribed system of points. Let a system of points \(t_1<\cdots<t_m\) be given. The matrices \(B\) and \(A\) considered above determine the class \(F\) of all real functions \(f(t)\) satisfying the conditions \(|f(t_\nu)-f(t_k)|\le b_{\nu k}\) \((\nu,k=1,\ldots,m)\), and the class \(G\) of all real functions \(g(t)\) satisfying the conditions \(|g(t_\nu)-g(t_k)|\le a_{\nu k}\) \((\nu,k=1,\ldots,m)\). If we put \(\rho(f,g)=\max_{1\le k\le m}|f(t_k)-g(t_k)|\), then in this case the quantity \(\mathscr{E}_{G}(F)\) will coincide with the right-hand side of (4).

2. Classes of functions with a prescribed majorant of the modulus of continuity. Consider now the space \(C\) of all functions \(f(t)\) continuous on the interval \([a,b]\), where \(\rho(f,g)=\sup_{a\le t\le b}|f(t)-g(t)|\). Let \(\omega(u)\) be an arbitrary modulus of continuity, i.e., a continuous nondecreasing and subadditive function equal to zero for \(u=0\) (see (7)). By \(H_\omega\), as usual, denote the set of all real continuous functions \(f(t)\) satisfying the condition \(|f(t)-f(\tau)|\le \omega(|t-\tau|)\) for any two points \(t\) and \(\tau\) in \([a,b]\). In view of what was said in item 4, whatever the moduli of continuity \(\omega_1(u)\) and \(\omega_2(u)\), for any system of points \(a=t_1<\cdots<t_{m+1}=b\) and for any function \(f(t)\) for which \(|f(t_\nu)-f(t_k)|\le \omega_1(|t_\nu-t_k|)\), there exists a function \(g(t)\) such that \(|g(t_\nu)-g(t_k)|\le \omega_2(|t_\nu-t_k|)\) and

\[ \max_{1\le k\le m+1}|f(t_k)-g(t_k)| \le \frac{1}{2}\max_{0\le u\le b-a}\{\omega_1(u)-\omega_2(u)\}. \tag{5} \]

Successively supplementing the system of points \(t_k\) in the corresponding way (for example, choosing \(m=2^p\) and \(t_k=a+\dfrac{k-1}{2^p}(b-a)\)) and applying the diagonal process, after passing to the limit, by compactness of the class \(H_{\omega_2}\), we obtain from this that there exists a function \(g_*(t)\in H_{\omega_2}\) satisfying the inequality

\[ \rho(f,g_*)\le \frac{1}{2}\max_{0\le u\le b-a}\{\omega_1(u)-\omega_2(u)\}. \tag{6} \]

The last inequality cannot be improved.

Thus, the following holds.

Theorem 1. Whatever the moduli of continuity \(\omega_1(u)\), \(\omega_2(u)\) in the space \(C\) of functions continuous on the interval \([a,b]\), the equality

\[ \mathscr{E}^{C}_{H_{\omega_2}}(H_{\omega_1}) =\frac{1}{2}\max_{0\le u\le b-a}\{\omega_1(u)-\omega_2(u)\} \tag{7} \]

always holds.

Considering the space \(C^*\) of all continuous \(2\pi\)-periodic functions and the corresponding classes \(H^*_{\omega_1}\) and \(H^*_{\omega_2}\), from the same considerations we obtain:

Theorem 2. Whatever the moduli of continuity \(\omega_1(u)\), \(\omega_2(u)\), in the space \(C^*\) the equality

\[ \mathscr{E}^{C^*}_{H^*_{\omega_2}}\left(H^*_{\omega_1}\right) = \frac12 \max_{0\le u\le \pi}\{\omega_1(u)-\omega_2(u)\}. \tag{8} \]

always holds.

If, for example, \(\omega_1(t)=Mt^\alpha\) \((0<\alpha\le 1)\), and \(\omega_2(t)=Kt^\alpha\) \((K<M)\), then

\[ \mathscr{E}^{C^*}_{H^*_{\omega_2}}\left(H^*_{\omega_1}\right) = \frac{\pi^\alpha}{2}(M-K). \]

Theorem 3. Let \(Z_1\) be the class of all functions \(f(t)\) defined on \([-1,1]\) and satisfying the conditions \(f(+1)=f(-1)=0\), \(|f(t_1)-f(t_2)|\le 1\),

\[ \left|f(t_1)-2f\left(\frac{t_1+t_2}{2}\right)+f(t_2)\right|\le |t_1-t_2|,\qquad t_1,t_2\in[-1,1]. \]

Then in the space \(C\) of all functions continuous on \([-1,1]\), for \(\omega(u)=mu\) (\(m\) an integer), the equality

\[ \mathscr{E}^{C}_{H\omega}(Z_1)=\frac{1}{2^{m+1}} \]

holds.

The proof may be obtained with the aid of Theorem 1 from \((^6)\).

6. One application. For applications of equalities (7) and (8) the following obvious relation is useful:

\[ \inf_M\left\{\max_{0\le u\le b-a}\bigl[\omega_1(u)-M\omega_2(u)\bigr]+M\omega_2(h)\right\} = \omega_1(h), \tag{9} \]

valid for any positive \(h\le b-a\), under the condition that \(\omega_1(u)\) and \(\omega_2(u)\) are arbitrary convex moduli of continuity for which, for every \(M\ge 0\), the difference \(\omega_1(u)-M\omega_2(u)\) is either monotone on \([0,b-a]\), or has an increment which, as \(u\) increases, changes sign once from plus to minus. In particular, equality (9) is always valid for \(\omega_2(u)=u\). An example of Cantor’s singular function, which is a modulus of continuity (see \((^7)\), § 3.2.4), shows that even in the case \(\omega_2(u)=u\), without the assumption of convexity of the function \(\omega_1(u)\), relation (9) may already fail.

Let us note one such application, the idea of which belongs to Lebesgue \((^9)\) (see also \((^2)\), p. 238) and occurs in \((^1)\) (p. 215). By virtue of equality (8) for \(\omega_2(u)=Mu\) and relation (9) for \(h=\pi/n\), relying on the well-known result of N. I. Akhiezer—M. G. Krein and J. Favard \((^7)\), we have that the best uniform approximation by trigonometric polynomials of order \(\le n-1\) of any \(2\pi\)-periodic function \(f(t)\in H^*_{\omega_1}\) does not exceed \(\frac12\omega_1(\pi/n)\). The latter estimate, as the example of the \(2\pi/n\)-periodic function

\[ f(t)= \begin{cases} \omega_1(t), & 0\le t\le \dfrac{\pi}{n},\\[6pt] \omega_1\left(\dfrac{2\pi}{n}-t\right), & \dfrac{\pi}{n}\le t\le \dfrac{2\pi}{n}, \end{cases} \tag{10} \]

shows, in the general case cannot be improved. In the case \(\omega_1(u)=u^\alpha\) \((0<\alpha<1)\), N. P. Korneichuk arrived at it, relying on considerations connected with the theorems of P. L. Chebyshev and Vallee-Poussin on alternation.*

In view of the fact that the function (10) depends on \(n\), we give the following proposition:

Theorem 4. For any convex modulus of continuity \(\omega_1(u)\)

\[ \sup_{f\in H^*_{\omega_1}}\varlimsup_{n\to\infty} \frac{E^*_{n-1}(f)}{\omega_1\left(\dfrac{\pi}{n}\right)} = \frac12. \tag{11} \]

* Communication at the seminar on function theory at Dnepropetrovsk University, March 21 and 28, 1961.

where

\[ E_{n-1}^{*}(f)=\inf_{a_k,b_k}\max_t \left|f(t)-\sum_{k=0}^{n-1}(a_k\cos kt+b_k\sin kt)\right|. \]

For the proof of Theorem 4 one may use a result of S. M. Nikol’skii\(^5\) on the upper bound for the approximation of functions of the class \(H_{\omega}^{*}\) by interpolating trigonometric polynomials with equidistant nodes.

Theorem 5. If \(\omega_1(u)\) is an arbitrary convex modulus of continuity, and \(D_{2n-1}(H_{\omega_1}^{*})\) is the width of order \(2n-1\) (see \((^3,^8)\)) for \(H_{\omega_1}^{*}\), then

\[ D_{2n-1}(H_{\omega_1}^{*})=\frac12\,\omega_1\left(\frac{\pi}{n}\right). \tag{12} \]

This theorem supplements a recent result of Lorentz\(^4\).

Dnepropetrovsk State University
named for the 300th anniversary of the reunification of Ukraine with Russia

Received
27 IV 1961

References

\(^1\) N. I. Akhiezer, Lectures on the Theory of Approximation, 1947.
\(^2\) V. L. Goncharov, Theory of Interpolation and Approximation of Functions, 1954.
\(^3\) A. N. Kolmogorov, Ann. of Math., 37, 107 (1936).
\(^4\) G. G. Lorentz, Bull. Am. Math. Soc., 66, No. 2, 124 (1960).
\(^5\) S. M. Nikol’skii, Tr. Mat. Inst. im. V. A. Steklova, Akad. Nauk SSSR, 15 (1945).
\(^6\) A. F. Timan, Izv. Akad. Nauk SSSR, Ser. Mat., 15, No. 3 (1951).
\(^7\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, 1960.
\(^8\) V. M. Tikhomirov, DAN, 130, No. 4, 734 (1960); UMN, No. 3 (1960).
\(^9\) H. Lebesgue, Bull. Sci. Math., 22 (1898).