Mathematics

N. P. KORNEICHUK

ON THE BEST UNIFORM APPROXIMATION

ON CERTAIN CLASSES OF CONTINUOUS FUNCTIONS

(Presented by Academician A. N. Kolmogorov, 12 V 1961)

1°. We shall denote by \(K H^{(\alpha)}\) the class of functions \(f(t)\) of period \(2\pi\) satisfying, on the entire real axis, the Lipschitz condition

\[ |f(t')-f(t'')|\leq K|t'-t''|^\alpha,\qquad (0<\alpha\leq 1). \tag{1} \]

The class of functions \(f\) satisfying condition (1) on the interval \([-1,1]\) will be denoted by \(K H^{(\alpha)}_{[-1,1]}\).

Let \(E_n(f)\) be the best uniform approximation of a periodic function \(f\) by trigonometric polynomials of order not exceeding \(n\); \(E_n(f;-1,1)\) the best approximation of the function \(f\) by algebraic polynomials of degree not exceeding \(n\) on the interval \([-1,1]\).

Favard \((^1)\) and, simultaneously with him, N. I. Akhiezer and M. G. Krein \((^2)\) obtained exact expressions for the upper bounds of the best approximations of periodic functions having a bounded derivative of order \(r\). In particular, they showed that

\[ \sup_{f\in K H^{(1)}} E_n(f)=\frac{K\pi}{2(n+1)},\qquad n=0,1,2,\ldots . \tag{2} \]

For the case \(0<\alpha<1\), Favard indicated the obvious lower estimate:

\[ \sup_{f\in K H^{(\alpha)}} E_n(f)\geq \frac{K\pi^\alpha}{2(n+1)^\alpha}. \tag{3} \]

S. M. Nikol’skii \((^3)\) established that

\[ \sup_{f\in K H^{(1)}_{[-1,1]}} E_n(f;-1,1) =\frac{K\pi}{2(n+1)}-\varepsilon_n, \]

where \(\varepsilon_n>0,\ \varepsilon_n=O\!\left(\dfrac{1}{n\lg n}\right)\).

Generalizing this result, S. N. Bernstein \((^4)\) showed that, if (without loss of generality one may put \(K=1\))

\[ \sup_{f\in H^{(\alpha)}_{[-1,1]}} E_n(f;-1,1) =\frac{c_n(\alpha)}{(n+1)^\alpha},\qquad \sup_{f\in H^\alpha} E_n(f) =\frac{\widetilde c_n(\alpha)}{(n+1)^\alpha}, \qquad n=0,1,2,\ldots, \]

then for \(0<\alpha\leq 1\)

\[ c_n(\alpha)\leq \widetilde c_n(\alpha)\leq \lim_{n\to\infty} c_n(\alpha) = \lim_{n\to\infty} \widetilde c_n(\alpha) = c(\alpha). \tag{4} \]

From (3), or from the fact that \(c_0(\alpha)=\pi^\alpha/2\), in view of (4) it followed that \(c(\alpha)\geq \pi^\alpha/2\). The exact value of the constant \(c(\alpha)\) for \(0<\alpha<1\) remained unknown.

We shall prove that \(c(\alpha)=\pi^\alpha/2\). This will immediately follow from the following assertion:

Theorem. For all \(n=0,1,2,\ldots\) and \(0<\alpha\leqslant 1\),
\[ \sup_{f\in H^{(\alpha)}} E_n(f)=\frac{\pi^\alpha}{2(n+1)^\alpha}. \tag{5} \]

Taking (2) and (3) into account, it is enough to establish that, for \(0<\alpha<1\),
\[ \sup_{f\in H^{(\alpha)}} E_n(f)\leqslant \frac{\pi^\alpha}{2(n+1)^\alpha}. \tag{6} \]

The proof of inequality (6) is based on the following lemmas.

Lemma 1. Let \(f(t)\in H^{(\alpha)}\), and let \(f_1(t)\) be a polygonal line inscribed in \(f\) in an arbitrary way. Then \(f_1\in H^{(\alpha)}\).

The proof of Lemma 1, based on the fact that the function \(t^\alpha\) is concave downward for \(t>0\), presents no special difficulties.

Let us consider periodic functions satisfying condition (1) on the whole axis and whose graph is a polygonal line with a finite number of vertices on the period \([0,2\pi)\). We shall denote the class of such functions by \(Kh^{(\alpha)}\).

Lemma 2. For all \(n=0,1,2,\ldots,\ 0<\alpha\leqslant 1\),
\[ \sup_{f\in H^{(\alpha)}} E_n(f)=\sup_{f\in h^{(\alpha)}} E_n(f). \tag{7} \]

Since \(h^{(\alpha)}\subset H^{(\alpha)}\), we have
\[ \sup_{f\in h^\alpha} E_n(f)\leqslant \sup_{f\in H^{(\alpha)}} E_n(f). \tag{8} \]

On the other hand, let \(f\in H^{(\alpha)}\), and let \(T_n(f,t)\) be the trigonometric polynomial of best approximation to the function \(f\). Then on \([0,2\pi)\) there are \(2n+2\) points \(x_1,x_2,\ldots,x_{2n+2}\) at which the difference \(f-T_n\) takes the values \(\pm E_n(f)\), alternately changing sign. Define a continuous periodic function \(f_1(t)\) by setting \(f_1(x_k)=f(x_k)\) and taking \(f_1(t)\) to be linear on the intervals \((x_k,x_{k+1})\). By Lemma 1, \(f_1\leqslant h^{(\alpha)}\), and the well-known theorem of Valle-Poussin (see, for example, [5]) allows us to conclude that \(E_n(f_1)\geqslant E_n(f)\), and consequently
\[ \sup_{f\in h^{(\alpha)}} E_n(f)\geqslant \sup_{f\in H^{(\alpha)}} E_n(f). \tag{9} \]

From (8) and (9), (7) follows.

Lemma 3. For arbitrary \(K>0,\ 0<\alpha<1\), for every function \(f\in h^{(\alpha)}\) there exists a function \(\varphi_0\in Kh^{(1)}\) such that
\[ \|f-\varphi_0\|=\max_t |f(t)-\varphi_0(t)| \leqslant \frac{1-\alpha}{2}\left(\frac{K}{\alpha}\right)^{\frac{\alpha}{\alpha-1}}. \tag{10} \]

If \(f\in Kh^{(1)}\), then the lemma is trivial; therefore we assume that \(f\notin Kh^{(1)}\), i.e. the polygonal line \(f(t)\) has at least one segment whose angular coefficient is, in absolute value, greater than \(K\).

Let \(x_1,x_2,\ldots,x_q\) be the abscissas of the vertices of the polygonal line \(f\) on \([0,2\pi)\), and let \(M\) be the set of all functions of the class \(Kh^{(1)}\) having, on \([0,2\pi]\), vertices at the points with abscissas \(x_k,\ k=1,2,\ldots,q\), and only at these points. Let \(\varphi_0\) be that function in \(M\) for which
\[ \|f-\varphi_0\|=\inf_{\varphi\in M}\|f-\varphi\|=\rho, \tag{11} \]
and, since \(f\notin Kh^{(1)}\), \(\rho>0\). The validity of Lemma 3 will be estab-

established if we show that

\[ \rho \leq \frac{1-\alpha}{2}\left(\frac{K}{\alpha}\right)^{\frac{\alpha}{\alpha-1}} . \]

The proof of this inequality is based on the following circumstance: among the links of the broken line \(\varphi_0\) there is at least one link \(l\), with abscissas of the endpoints \(x_k\) and \(x_m\), satisfying the following two conditions: 1) the angular coefficient \(\gamma\) of the link \(l\) is equal in absolute value to \(K\); 2) \(\varphi_0(x_k)-f(x_k)=f(x_m)-\varphi_0(x_m)=\rho\,\operatorname{sign}\gamma\).

Indeed, assuming that the broken line \(\varphi_0\) has no such link, one can show the possibility of constructing a function \(\varphi_1\in M\) for which \(\|f-\varphi_1\|<\rho\), which is impossible in view of (11).

But if for the link \(l\) on the interval \((x_k,x_m)\) conditions 1) and 2) are fulfilled, then

\[ |f(x_k)-f(x_m)|=2\rho+K|x_k-x_m|. \]

Hence

\[ \rho \leq \frac{1}{2}\{|x_k-x_m|^\alpha-K|x_k-x_m|\} \leq \frac{1}{2}\max_{t>0}(t^\alpha-Kt) = \frac{1-\alpha}{2}\left(\frac{K}{\alpha}\right)^{\frac{\alpha}{\alpha-1}} . \]

Now it is easy to prove (6). Let \(f\in h^{(\alpha)}\), and for any fixed \(K>0\), let \(\varphi_0\) be that function from the class \(Kh^{(\alpha)}\) for which (10) holds. If \(T_n(\varphi_0,t)\) is the trigonometric polynomial of best approximation for the function \(\varphi_0\in Kh^{(1)}\), then, by (2),

\[ E_n(t)\leq \|f(t)-T_n(\varphi_0,t)\| \leq \|f-\varphi_0\|+\|\varphi_0-T_n(\varphi_0,t)\|\leq \]

\[ \leq \frac{1-\alpha}{2}\left(\frac{K}{\alpha}\right)^{\frac{\alpha}{\alpha-1}} +\frac{K\pi}{2(n+1)}=\psi(K). \]

Since this is valid for every \(f\in h^{(\alpha)}\) and arbitrary \(K>0\), in view of (7),

\[ \sup_{f\in H^{(\alpha)}} E_n(f) = \sup_{f\in h^{(\alpha)}} E_n(l) \leq \min_{K>0}\psi(K) = \frac{1}{2}\left(\frac{\pi}{n+1}\right)^\alpha . \]

\(2^\circ\). Result (5) generalizes to the class \(H_\omega\) of continuous periodic functions whose modulus of continuity does not exceed a given modulus of continuity \(\omega(t)\), under the additional assumption that the function \(\omega(t)\) is convex upward for \(t>0\). In this case the same method of reasoning makes it possible to establish that

\[ \sup_{f\in H_\omega} E_n(f)=\frac{1}{2}\omega\left(\frac{\pi}{n+1}\right). \tag{12} \]

Indeed, Lemmas 1 and 2 are formulated and proved in a completely analogous way; \(h_\omega\) is defined as the subclass of functions of the class \(H_\omega\) which are broken lines with a finite number of nodes on the period. Then in Lemma 3 one may immediately choose \(K\) satisfying the inequalities
\[ \omega_+'\left(\frac{\pi}{n+1}\right)\leq K\leq \omega_-'\left(\frac{\pi}{n+1}\right), \]
and, using the fact that for such a choice of \(K\)
\[ \max_{t>0}[\omega(t)-Kt] = \omega\left(\frac{\pi}{n+1}\right)-K\frac{\pi}{n+1}, \]
by the same method prove the existence of a function \(\varphi_0\in Kh^{(1)}\) for which
\[ \|f-\varphi_0\|\leq \frac{1}{2}\left\{\omega\left(\frac{\pi}{n+1}\right)-K\frac{\pi}{n+1}\right\}^{*}. \]

\[ \overline{\phantom{xxxxx}} \]

* Since into every function \(f\in H_\omega\) one can inscribe \(f_1\in h_\omega\) so that \(\|f-f_1\|<\varepsilon\), the preceding arguments make it easy to conclude that
\[ \sup_{f\in H_\omega}\inf_{\varphi\in KH^{(1)}}\|f-\varphi\| = \frac{1}{2}\max_{t>0}[\omega(t)-Kt]. \]

3°. Let \(U_n\) be a linear operator by means of which to each periodic integrable function \(f\) there is assigned the trigonometric polynomial

\[ U_n(f,x,\lambda)=\frac{a_0}{2}+\sum_{k=1}^{n}\lambda_k^{(n)}(a_k\cos kx+b_k\sin kx), \tag{13} \]

where \(a_0,\ a_k,\ b_k,\ k=1,2,\ldots,n\), are the Fourier coefficients of the function \(f\), and \(\lambda_k^{(n)}\) are material multipliers.

If \(f_0\in H^{(\alpha)}\) is an even function with period \(\dfrac{2\pi}{n+1}\), defined on \(\left[0,\dfrac{\pi}{n+1}\right]\) by the equality \(f_0(t)=t^\alpha\), then, expanding \(f_0\) in a Fourier series, it is easy to compute that, whatever the \(\lambda_k^{(n)}\) may be, for all \(n=0,1,2,\ldots\),

\[ \sup_{f\in H^{(\alpha)}}\|f-U_nf\|\ge |U_n(f_0,0,\lambda)| =\frac{\pi^\alpha}{(1+\alpha)(n+1)^\alpha}\qquad (0<\alpha\le 1). \]

This, in comparison with (5), shows that for \(0<\alpha\le 1\), for every linear operator of the form (13),

\[ \sup_{f\in H^{(\alpha)}}\|f-U_nf\|\ge \sup_{f\in H^{(\alpha)}}E_n(f)+\frac{1-\alpha}{1+\alpha} \left(\frac{\pi}{n+1}\right)^\alpha . \]

By means of analogous reasoning it is easily proved that if the function \(\omega(t)\) is convex upward and is not linear on the interval \(\left(0,\dfrac{\pi}{n+1}\right)\), then

\[ \inf_{\lambda}\sup_{f\in H_\omega}\|f-U_nf\|\ge \frac{n+1}{\pi}\int_{0}^{\frac{\pi}{n+1}}\omega(t)\,dt > \frac{1}{2}\omega\left(\frac{\pi}{n+1}\right) = \sup_{f\in H_\omega}E_n(f), \]

i.e., there does not exist a linear operator (13) which would realize on the class \(H_\omega\) the upper bound of the best approximations*.

The author expresses his gratitude to S. M. Nikolskii for posing the problem and for his attention to the work.

Dnepropetrovsk State University

Received
10 IV 1961

CITED LITERATURE

\({}^{1}\) J. Favard, Bull. de Sci. Math., 61, 209 (1937).
\({}^{2}\) N. I. Akhiezer, M. G. Krein, DAN, 15, No. 3, 107 (1937).
\({}^{3}\) S. M. Nikolskii, Izv. AN SSSR, ser. matem., 10, No. 4, 295 (1946).
\({}^{4}\) S. N. Bernstein, DAN, 53, No. 7, 587 (1946).
\({}^{5}\) I. P. Natanson, Constructive Theory of Functions, 1949.

* The results presented in the article were reported by the author in final form on 10 III 1961 at the seminar of Dnepropetrovsk State University directed by M. I. Alkhimov. Preliminary results were reported in February 1961 at the university’s final scientific conference.