MATHEMATICS

N. P. KORNEICHUK

THE EXACT VALUE OF BEST APPROXIMATIONS AND WIDTHS OF CERTAIN CLASSES OF FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 14 I 1963)

1. Let \(C_{2\pi}\) be the space of continuous functions of period \(2\pi\) with norm
\(\|f\|=\max_x |f(x)|\), and let \(E_n(f)\) be the best uniform approximation of the function \(f\) by trigonometric polynomials of order not exceeding \(n\). If \(\mathfrak M\) is some set of functions, then we shall put

\[ E_n(\mathfrak M)=\sup_{f\in \mathfrak M} E_n(f). \]

Denote by \(W^{(r)}H_\omega\) \((r=0,1,2,\ldots)\) the class of functions \(f\in C_{2\pi}\) for which the derivative of order \(r\) has modulus of continuity \(\omega(f^{(r)};t)\) not exceeding a given modulus of continuity \(\omega(t)\). For \(\omega(t)=K t^\alpha\) \((0\le t\le \pi,\ 0<\alpha\le 1)\) we shall write \(W^{(r)}KH^{(\alpha)}\).

In the papers \((^{1,2})\) we gave the solution of Favard’s problem \((^3)\) for the classes \(W^{(0)}H_\omega=H_\omega\) and \(W^{(1)}H_\omega\), when \(\omega(t)\) is a convex function; namely, it was shown that under this condition

\[ E_{n-1}(H_\omega)=\frac12\,\omega\!\left(\frac{\pi}{n}\right) \qquad (n=1,2,\ldots); \tag{1} \]

\[ E_{n-1}\bigl(W^{(1)}H_\omega\bigr) = \frac14\int_0^{\pi/n}\omega(t)\,dt \qquad (n=1,2,\ldots). \tag{2} \]

To prove the equalities (1) and (2), we first obtained an exact estimate of the best approximations of functions of the classes \(H_\omega\) and \(W^{(1)}H_\omega\) by functions of the classes \(KH^{(1)}\) and \(W^{(1)}KH^{(1)}\), respectively, which then, for a suitably chosen (extremal) \(K\), were approximated by trigonometric polynomials. In doing so, the known \((^{3,4})\) equality was used

\[ E_{n-1}\bigl(W^{(r)}KH^{(1)}\bigr) = \frac{4K}{\pi n^{r+1}} \sum_{m=0}^{\infty} \frac{(-1)^{mr}}{(2m+1)^{r+2}} \qquad (n=1,2,\ldots;\ r=0,1,2,\ldots). \tag{3} \]

This method of ours, which led to the exact constant in the estimate of the quantities
\(E_{n-1}(W^{(r)}H_\omega)\) \((n=1,2,\ldots;\ r=0,1)\), makes it possible (under the condition of convexity of \(\omega(t)\)) to obtain the same result in the case \(r=2,3\). For the widths \((^{5,6})\) of the classes under consideration this gives an upper estimate which, as it was possible to show with the aid of a theorem of A. N. Kolmogorov, turned out to be exact.

2. Theorem 1. Let \(\omega(t)\) be an arbitrary modulus of continuity and let \(K\) be any positive number. If \(f\in W^{(2)}H_\omega\), then

\[ \inf_{\varphi\in W^{(2)}KH^{(1)}} \|f-\varphi\| \le \frac18\max_{x\ge 0}\int_0^x t\,|\omega(t)-Kt|\,dt. \tag{4} \]

Theorem 2. If \(\omega(t)\) is a convex modulus of continuity, then for all \(n=1,2,\ldots\) there exists a positive number \(K_n=K(n)\) such that,

that

\[ \sup_{f\in W^{(2)}H_\omega}\ \inf_{\varphi\in W^{(2)}K_nH^{(1)}} \|f-\varphi\| = \frac18 \int_0^{\pi/n} t\,|\omega(t)-K_n t|\,dt . \tag{5} \]

The proof of Theorems 1 and 2 is based on the following lemma.

Lemma. Let the function \(f\) have a second derivative \(f''\) belonging to the space \(C_{2\pi}\). Suppose further that a number \(K>0\) is given such that \(f''\in KH^{(1)}\). Then, in order that for a function \(\varphi_0\in W^{(2)}KH^{(1)}\) the equality

\[ \|f-\varphi_0\|= \inf_{\varphi\in W^{(2)}KH^{(1)}}\|f-\varphi\|=\rho \]

hold, it is necessary that there exist at least two points \(a\) and \(b\), \(a<b\), at which the following conditions are fulfilled:

\[ \begin{aligned} &1)\quad f(a)-\varphi_0(a)=\varphi_0(b)-f(b)=\pm\rho;\\ &2)\quad \varphi_0'(b)-\varphi_0'(a)=K(b-a)\operatorname{sign}[f(a)-\varphi_0(a)]. \end{aligned} \]

With the aid of Theorem 2 and equality (3) one proves

Theorem 3. If \(\omega(t)\) is a convex modulus of continuity, then

\[ E_{n-1}\bigl(W^{(2)}H_\omega\bigr) = \frac18 \int_0^{\pi/n} t\omega(t)\,dt \qquad (n=1,2,\ldots). \tag{6} \]

Corollary.

\[ E_{n-1}\bigl(W^{(2)}KH^{(\alpha)}\bigr) = \frac{K\pi^{2+\alpha}}{8(2+\alpha)n^{2+\alpha}} \qquad (n=1,2,\ldots,\ 0<\alpha<1). \]

It is not difficult to indicate in the class \(W^{(2)}H_\omega\) a function \(f_{n2}(x)\) whose best approximation \(E_{n-1}(f_{n2})\) is exactly equal to the right-hand side of (6). Let, for \(n=1,2,\ldots;\ r=0,1,2,\ldots\), \(f_{nr}(x)=f_{nr}(\omega;x)\) be a function of period \(2\pi/n\), with mean value over the period equal to zero, for which the derivative of order \(r\) is defined by the equalities

\[ f_{nr}^{(r)}(x)= \begin{cases} \dfrac12\,\omega(2x), & \left(0\le x\le \dfrac{\pi}{2n}\right),\\[6pt] \dfrac12\,\omega\!\left(\dfrac{2\pi}{n}-2x\right), & \left(\dfrac{\pi}{2n}\le x\le \dfrac{\pi}{n}\right), \end{cases} \]

\[ f_{nr}^{(r)}(x)=-f_{nr}^{(r)}(-x). \]

It is easy to verify that for convex \(\omega(t)\), \(f_{nr}\in W^{(r)}H_\omega\) and \(E_{n-1}(f_{nr})=\|f_{nr}\|\), and in the case \(r=2\), \(\|f_{nr}\|\) coincides with the right-hand side of (6).

Results analogous to Theorems 1, 2, and 3 were also obtained by us for the case \(r=3\). With the aid of the functions \(f_{nr}\), relations (1), (2), (6), and also the corresponding equality for the class \(W^{(3)}H_\omega\), can be written in the form

\[ E_{n-1}\bigl(W^{(r)}H_\omega\bigr)=\|f_{nr}\| \qquad (n=1,2,\ldots;\ r=0,1,2,3), \tag{7} \]

under the assumption that the modulus of continuity \(\omega(t)\) is convex. We note that for \(n=1\) equality (7) was established \(\bigl({}^{7}\bigr)\) for all \(r=0,1,2,\ldots\).

In the case when \(\omega(t)\) is not a convex function, our estimates of the quantities \(E_{n-1}(W^{(r)}H_\omega)\) are less precise. We give some results for \(r=2\). Put

\[ \eta(x,K)=\int_0^x t\,|\omega(t)-Kt|\,dt \]

and let \(X_\omega\) be the set of points \(x_0\) of the half-line \([0,+\infty)\) for which, for some \(K_0=K(x_0)\), the equality

\[ \max_{x\ge 0}\eta(x,K_0)=\eta(x_0,K_0) \tag{8} \]

holds.

From Theorem 1 and equality (3), for arbitrary \(\omega(t)\) we obtain the estimate

\[ E_{n-1}\left(W^{(2)}H_\omega\right)\leqslant \frac{1}{8}\left\{\int_0^{\pi/n}t\omega(t)\,dt+ \int_{\pi/n}^{x_0}t\,[\omega(t)-K_0t]\,dt\right\}, \tag{9} \]

where \(x_0\) is any point of the set \(X_0\), and the number \(K_0\) is related to \(x_0\) by equality (8).

If for the modulus of continuity \(\omega(t)\) there exist natural numbers
\(n_1<n_2<n_3<\cdots\) such that \(\pi/n_k\in X_0\), then

\[ E_{n_k-1}\left(W^{(2)}H_\omega\right)\leqslant \frac{1}{8}\int_0^{\pi/n_k}t\omega(t)\,dt \qquad (k=1,2,\ldots). \]

If \(\pi/n\notin X_0\), then naturally in estimate (9) one chooses \(x_0\in X_0\) so that the second integral assumes the least value.

1. Let \(d_n(\mathfrak M)\) be the \(n\)-th width of the set \(\mathfrak M\) in the space \(C_{2\pi}\), i.e., the lower bound of the deviations of the set \(\mathfrak M\) from all possible manifolds of dimension \(\leqslant n\). The notion of width for functional classes was introduced by A. N. Kolmogorov\({}^{5}\). V. M. Tikhomirov, approximating functions of the class \(H_\omega\) by piecewise constant functions, showed (see Theorem 3 in \({}^{6}\) and the correction in the abstract of that paper (RZhMat, 1962, 4B69)) that, for a convex modulus of continuity \(\omega(t)\),

\[ d_{2n-1}(H_\omega)=\frac{1}{2}\omega\left(\frac{\pi}{n}\right)=\|f_{n0}\| \qquad (n=1,2,\ldots). \tag{10} \]

From equality (7) it follows that

\[ d_{2n-1}\left(W^{(r)}H_\omega\right)\leqslant \|f_{nr}\| \qquad (n=1,2,\ldots;\ r=1,2,3). \]

To obtain a lower estimate, take on \([0,2\pi)\) a system of \(2n\) points
\(x_k=k\pi/n\) \((k=0,1,2,\ldots,2n-1)\). It can be shown that, for fixed \(n=1,2,\ldots\), whatever the system of indices
\(i_0,i_1,\ldots,i_{2n-1}\), \(i_k=\pm1\), in each of the classes
\(W^{(r)}H_\omega\) \((r=1,2,3)\), where \(\omega(t)\) is a convex modulus of continuity, there exists a function \(f_r(x)\) such that
\(f_r(x_k)\geqslant \|f_{nr}\|\) if \(i_k=+1\), and
\(f_r(x_k)\leqslant -\|f_{nr}\|\) if \(i_k=-1\).

By virtue of A. N. Kolmogorov’s theorem (see \({}^{6}\), Theorem 2), from this we conclude that

\[ d_{2n-1}\left(W^{(r)}H_\omega\right)\geqslant \|f_{nr}\| \qquad (n=1,2,\ldots;\ r=1,2,3). \]

Thus, the following holds.

Theorem 4. If \(\omega(t)\) is a convex modulus of continuity, then

\[ d_{2n-1}\left(W^{(r)}H_\omega\right)=\|f_{nr}\| \qquad (n=1,2,\ldots;\ r=1,2,3). \tag{11} \]

From equalities (7), (10), and (11) it follows that, for each \(n=1,2,\ldots\), the set of trigonometric polynomials of degree not exceeding \(n-1\) is an extremal subspace of dimension \(2n-1\) for the classes
\(W^{(r)}H_\omega\) \((r=0,1,2,3)\) when \(\omega(t)\) is convex.

Received
10 I 1963

REFERENCES

\({}^{1}\) N. P. Korneichuk, DAN, 140, No. 4, 748 (1961).
\({}^{2}\) N. P. Korneichuk, DAN, 141, No. 2, 304 (1961).
\({}^{3}\) J. Favard, Bull. Sci. Math., 61, 209 (1937).
\({}^{4}\) N. I. Akhiezer, M. G. Krein, DAN, 15, No. 2, 107 (1937).
\({}^{5}\) A. Kolmogoroff, Ann. of. Math., 37, No. 1, 107 (1936).
\({}^{6}\) V. M. Tikhomirov, UMN, 15, issue 3 (93), 81 (1960).
\({}^{7}\) N. P. Korneichuk, Dokl. AN USSR, No. 8, 993 (1962).