MATHEMATICS

N. P. KORNEICHUK

THE EXACT CONSTANT IN D. JACKSON’S THEOREM ON BEST UNIFORM APPROXIMATION OF CONTINUOUS PERIODIC FUNCTIONS

(Presented by Academician P. S. Novikov, 27 II 1962)

1. Let \(C_{2\pi}\) be the space of continuous functions with period \(2\pi\). If the modulus of continuity \(\omega(f;t)\) of a function \(f\in C_{2\pi}\), for all \(0\leq t\leq \pi\), does not exceed some convex modulus of continuity \(\omega_1(t)\), then, as was shown in the paper \((^1)\), for the best uniform approximation \(E_n(f)\) of the function \(f\) by trigonometric polynomials of degree not exceeding \(n\), the estimate

\[ E_n(f)\leq \frac12\,\omega_1\!\left(\frac{\pi}{n+1}\right)\quad (n=0,1,2,\ldots). \tag{1} \]

holds.

By virtue of a lemma of S. B. Stechkin (see \((^2)\), Lemma 4), for any modulus of continuity \(\omega(t)\), \(\omega(t)\not\equiv 0\), there exists a convex monotone majorant \(\omega_1(t)\) such that \(\omega(t)\leq \omega_1(t)<2\omega(t)\). Hence from (1) it follows at once that, whatever the function \(f\in C_{2\pi}\), \(f\not\equiv \mathrm{const}\), one has

\[ E_n(f)<\omega\!\left(f;\frac{\pi}{n+1}\right)\quad (n=0,1,2,\ldots). \]

S. B. Stechkin conjectured that the last estimate cannot be improved on the whole space \(C_{2\pi}\).

We shall show that this is indeed so.

Lemma. For any fixed \(n=0,1,2,\ldots\), whatever the number \(\varepsilon>0\), one can specify a function \(\varphi\in C_{2\pi}\), not identically constant, such that

\[ E_n(\varphi)>\left(\frac{2n+1}{2n+2}-\varepsilon\right) \omega\!\left(\varphi;\frac{\pi}{n+1}\right). \tag{2} \]

Let \(\varepsilon>0\) be given, and we may assume \(0<\varepsilon<\tfrac12\). Put

\[ h=\frac{\pi}{n+1},\qquad x_0=0,\qquad x_k=kh-(n-k+1)\beta\quad (k=1,2,\ldots,n+1), \]

where the number \(\beta\) satisfies the inequalities

\[ 0<\beta<\frac{2\varepsilon}{(n+1)^2}, \]

and therefore

\[ x_0<x_1<x_2<\cdots<x_{n+1}=\pi. \]

Construct a continuous even function \(\varphi(x)\) with period \(2\pi\), defining it on the interval \([0,\pi]\) as follows:
\[ \varphi(x_k)=(-1)^{k+1}\quad (k=1,2,\ldots,n+1), \]
\[ \varphi(x)=0,\quad \text{if } 0\leq x\leq x_1-\beta \quad \text{and} \quad x_k+\beta\leq x\leq x_{k+1}-\beta\quad (k=1,2,\ldots,n+1), \]
and \(\varphi(x)\) is linear on the intervals
\[ [x_k-\beta,x_k]\quad (k=1,2,\ldots,n+1) \]
and
\[ [x_k,x_k+\beta]\quad (k=1,2,\ldots,n). \]

It is easy to verify that

\[ \omega\!\left(\varphi;\frac{\pi}{n+1}\right)=1. \]

Let

\[ T_n(x)=\frac{1}{n+1}\left(\frac12+\cos x+\cos 2x+\cdots+\cos nx\right) =\frac{1}{n+1}\,\frac{\sin (n+\tfrac12)x}{2\sin \tfrac12 x}. \]

As is not difficult to compute,

\[ T_n(x_0)=T_n(0)=\frac{2n+1}{2n+2},\qquad T_n(kh)=\frac{(-1)^{k+1}}{2n+2} \quad (k=1,2,\ldots,n+1), \]

\[ \left|T_n(x_k)-T_n(kh)\right| <\frac{n}{2}(n-k+1)\beta<\varepsilon \quad (k=1,2,\ldots,n). \]

Consequently,

\[ \varphi(x_k)-T_n(x_k)=[\varphi(x_k)-T_n(kh)]+[T_n(kh)-T_n(x_k)] \]

\[ =(-1)^{k+1}\frac{2n+1}{2n+2}+\mu_k \qquad (k=0,1,2,\ldots,n+1), \]

where \(0\leq |\mu_k|<\varepsilon<1/2\).

Taking into account the evenness of \(\varphi(x)\) and \(T_n(x)\), we see that the difference \(\varphi(x)-T_n(x)\) on the period \((-\pi,\pi]\) assumes, at \(2n+2\) points, values with alternating signs, and these values in absolute magnitude exceed \(\frac{2n+1}{2n+2}-\varepsilon\). Therefore

\[ E_n(\varphi)>\frac{2n+1}{2n+2}-\varepsilon =\left(\frac{2n+1}{2n+2}-\varepsilon\right)\omega\left(\varphi;\frac{\pi}{n+1}\right), \]

and the lemma is proved.

Thus, D. Jackson’s theorem (see, for example, \({}^{3}\), p. 238) for the space \(C_{2\pi}\) takes the following form.

Theorem. If \(f(x)\in C_{2\pi}\), then

\[ E_n(f)\leq \omega\left(f;\frac{\pi}{n+1}\right) \qquad (n=0,1,2,\ldots), \tag{3} \]

where equality in relation (3) occurs only in the case when \(f(x)\) is identically constant.

The absolute constant \(1\) on the right-hand side of inequality (3) is definitive*.

Let us note an obvious consequence.

Corollary. The limiting equality holds

\[ \lim_{n\to\infty}\ \sup_{f\in C_{2\pi}} \frac{E_n(f)}{\omega\left(f;\frac{\pi}{n+1}\right)}=1. \tag{4} \]

1. Let \(C_{[a,b]}\) be the space of functions continuous on the interval \([a,b]\), and let \(E_n(f;a,b)\) be the best uniform approximation of a function \(f\in C_{[a,b]}\) on this interval by algebraic polynomials of degree \(\leq n\).

Then, putting

\[ f\left[\frac{(b-a)\cos\theta+(a+b)}{2}\right]=\psi(\theta) \]

and taking into account that, if \(f\in C_{[a,b]}\), then \(\psi\in C_{2\pi}\) and, moreover (see, for example, \({}^{4}\)), \(E_n(f;a,b)=E_n(\psi)\), \(\omega(\psi;t)\leq \omega\left(f;\frac{b-a}{2}t\right)\), with the help of (3) we arrive at the estimate

\[ E_n(f;a,b)\leq \omega\left(f;\frac{b-a}{2}\,\frac{\pi}{n+1}\right) \qquad (n=0,1,2,\ldots), \]

valid for every function \(f\) of the space \(C_{[a,b]}\).

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

Received
27 II 1962

REFERENCES

\({}^{1}\) N. P. Korneichuk, DAN, 140, 748 (1961).
\({}^{2}\) A. V. Efimov, Matem. sbornik, 54 (96), No. 1, 51 (1961).
\({}^{3}\) V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Moscow, 1954.
\({}^{4}\) I. P. Natanson, Constructive Theory of Functions, Moscow–Leningrad, 1949.

* That is, inequality (3) ceases to be true simultaneously for all \(f\in C_{2\pi}\) and \(n=0,1,2,\ldots\), if its right-hand side is multiplied by \(1-\varepsilon\), where \(\varepsilon\) is a positive number independent of both \(f\) and \(n\).