MATHEMATICS

V. P. MOTORNYY

ON AN INEQUALITY FOR MODULI OF SMOOTHNESS OF A PERIODIC FUNCTION WITH BOUNDED DERIVATIVE

(Presented by Academician S. N. Bernstein on 20 VII 1963)

Let us consider, for some natural number \(p\), the class \(W_*^{(p)}\) of all periodic functions \(f(x)\) of period \(2\pi\) that have an absolutely continuous derivative of order \((p-1)\) and are such that \(\left|f^{(p)}(x)\right| \leqslant 1\) almost everywhere. An important place in this class is occupied by the well-known Bernoulli functions

\[ \varphi_p(x)=\frac{4}{\pi}\sum_{m=0}^{\infty} \frac{\sin [(2m+1)x-p\pi/2]}{(2m+1)^{p+1}}, \tag{1} \]

which arise in a number of extremal problems for periodic differentiable functions. The role of the functions (1) in such problems was first pointed out by S. N. Bernstein \(\left({}^{1}\right)\), who proved that if \(f(x)\in W_*^{(p)}\) and

\[ \int_0^{2\pi} f(x)\,dx=0, \tag{2} \]

then

\[ \max_x |f(x)| \leqslant \max_x \varphi_p(x). \tag{3} \]

The present note is devoted to the solution of one extremal problem in which the Bernoulli functions also occupy a central place.

Theorem 1. For any natural number \(k \leqslant p+1\) the equality

\[ \sup_{f\in W_*^{(p)}} \omega_k(f;t)=\omega_k(\varphi_p;t), \tag{4} \]

holds, where

\[ \omega_k(f;t)=\max_{x,|h|\leqslant t} \left|\sum_{\nu=0}^{k}(-1)^{k-\nu}\binom{k}{\nu} f(x+\nu h)\right| \]

is the \(k\)-th modulus of smoothness of the function \(f(x)\).

The proof of relation (4) is based on the known (see, for example, \(\left({}^{6}\right)\), p. 131) integral representation for functions \(f(x)\in W_*^{(p)}\):

\[ f(x)=\frac{1}{\pi}\int_0^{2\pi} f^{(p)}(t) \sum_{\nu=1}^{\infty} \frac{\cos [\nu(x-t)-p\pi/2]}{\nu^p}\,dt \tag{5} \]

and on the investigation of the sign of the differences of the kernel occurring under the integral in (5).

In the case \(k=1\), Theorem 1 follows easily from the work of A. F. Timan (see \(\left({}^{4}\right)\), Theorem 1).

For \(k=p\), Theorem 1 and a theorem of D. A. Raikov \(\left({}^{2}\right)\) imply the validity of the following assertion, earlier stated by A. F. Timan without proof at his seminar on the theory of functions.

Theorem 2. In order that a periodic function of period \(2\pi\) belong to the class \(W_*^{(p)}\), it is necessary and sufficient that, for every \(t \geqslant 0\), it satisfy the condition

\[ \omega_p(f;t)\leqslant \omega_p(\varphi_p;t). \tag{6} \]

We note that Theorem 1 loses its force for \(k>p+1\). This is seen from the following example.

Let \(k=3\) and \(p=1\). If we set

\[ f^{(1)}(\pi+x)= \begin{cases} -1, & 0\leq x\leq \frac14\pi,\\ 1, & \frac14\pi<x\leq \frac34\pi,\\ -1, & \frac34\pi<x\leq \frac54\pi,\\ 1, & \frac54\pi<x\leq \frac74\pi,\\ -1, & \frac74\pi<x\leq 2\pi, \end{cases} \]

then it is not hard to verify that, for \(\frac13\pi<t<\frac23\pi\),
\(\omega_3(f;t)>\omega_3(\varphi_1;t)\).

It can be shown that the value of the right-hand side of (4), for all nonnegative \(t\leq\pi\) and \(k\leq p+1\), is determined by the equality

\[ \omega_k(\varphi_p;t) = \frac{4}{\pi} \left| \sum_{\nu=0}^{\infty} (-1)^{\nu(p+k+1)} \frac{\{2\sin(2\nu+1)t/2\}^{k}}{(2\nu+1)^{p+1}} \right|. \tag{7} \]

Denote by \(\widetilde W_*^{(p)}\) the class of all functions \(f(x)\), periodic with period \(2\pi\), trigonometrically conjugate to functions from \(W_*^{(p)}\). Along with Theorem 1, we also note the following proposition.

Theorem 3. For any natural \(k\leq p\) the equality

\[ \sup_{f\in \widetilde W_*^{(p)}} \omega_k(f;t) = \omega_k(\widetilde\varphi_p;t), \]

holds, where \(\widetilde\varphi_p(x)\) is the function trigonometrically conjugate to \(\varphi_p(x)\).

If \(0\leq t\leq\pi\), then

\[ \omega_k(\widetilde\varphi_p;t) = \frac{4}{\pi} \left| \sum_{\nu=0}^{\infty} (-1)^{\nu(p+k)} \frac{\{\sin(2\nu+1)t/2\}^{k}}{(2\nu+1)^{p+1}} \right|. \tag{8} \]

From Theorem 3, for \(p=1,\ k=1\), there immediately follows a corollary sharpening a known theorem of I. I. Privalov (see, for example, \((^5)\), Ch. VII, p. 59), namely, that if

\[ |f(x)-f(x+h)|\leq h, \]

then

\[ |\widetilde f(x)-\widetilde f(x+h)| = O\!\left(h\ln\frac1h\right). \]

Corollary 1. If a periodic function \(f(x)\) of period \(2\pi\) satisfies the Lipschitz condition

\[ |f(x)-f(x+h)|\leq h, \]

then for the function \(\widetilde f(x)\), trigonometrically conjugate to \(f(x)\), for any nonnegative \(t\leq\pi\) the inequality

\[ \omega(\widetilde f;t) \leq \frac{2}{\pi}\,t\ln\operatorname{ctg}\frac{t}{4} + \frac{4}{\pi}\int_{0}^{t/2}\frac{x}{\sin x}\,dx, \tag{9} \]

holds, and it is sharp.

Let \(W_*^{(p)}(L)\) be the class of functions \(f(x)\), periodic with period \(2\pi\), having an absolutely continuous derivative of order \((p-1)\) and such that

\[ \int_{0}^{2\pi}|f^{(p)}(t)|\,dt\leq 1. \]

Let, further, \(\widetilde W_*^{(p)}(L)\) be the class of all functions \(f(x)\) of period \(2\pi\) trigonometric-conjugate to functions from \(W_*^{(p)}(L)\).

Then the following theorems hold:

Theorem 4. For any natural \(k \leq p+1\) the equality
\[ \sup_{f\in W_*^{(p)}(L)} \omega_k(f;t)_L=\omega_k(\varphi_p;t), \tag{10} \]
holds, where \(\omega_k(f;t)_L\) is the integral modulus of smoothness:
\[ \omega_k(f;t)_L=\sup_{|h|\leq t}\int_0^{2\pi} \left|\sum_{\nu=0}^{k}(-1)^{k-\nu}\binom{k}{\nu} f(x+\nu h)\right|\,dx. \]

Theorem 5. For natural \(k \leq p\) the equality
\[ \sup_{f\in \widetilde W_*^{(p)}(L)} \omega_k(f;t)_L=\omega_k(\varphi_p;t) \tag{11} \]
holds.

The validity of relations (10) and (11) follows from Theorems 1 and 3 and from an equality of S. M. Nikol’skii (see \({}^{3}\), Theorem 3):
\[ \sup_{f\in W_*^{(p)}(L)} \int_0^{2\pi}\left|\int_0^{2\pi} f^{(p)}(t)\sum_{\nu=0}^{\infty} \frac{\cos[\nu(x-t)-p\pi/2]}{\nu^p}\,dt\right|\,dx = \]
\[ =\frac12\max_t\int_0^{2\pi} \left|\sum_{\nu=0}^{\infty} \frac{\cos[\nu(x-t)-p\pi/2]-\cos[\nu x+p\pi/2]}{\nu^p}\right|\,dx. \]

From Theorem 5, for \(p=1,\ k=1\), the following corollary follows, refining a theorem analogous to a theorem of I. I. Privalov.

Corollary 2. If a periodic function \(f(x)\) of period \(2\pi\) satisfies the condition
\[ \int_0^{2\pi}|f(x)-f(x+t)|\,dx \leq t, \]
then for the function \(\widetilde f(x)\), trigonometric-conjugate to \(f(x)\), for any nonnegative \(t\leq \pi\) the inequality
\[ \omega(\widetilde f;t)_L \leq \frac{2}{\pi}\,t\ln\operatorname{ctg}\frac{t}{4} +\frac{4}{\pi}\int_0^{t/2}\frac{x}{\sin x}\,dx, \tag{12} \]
holds, and it is sharp.

In conclusion, I consider it my duty to express deep gratitude to Prof. A. F. Timan for posing the problem and for his constant attention.

Received
17 VII 1963

REFERENCES

\({}^{1}\) S. N. Bernstein, C. R., 200, 1900 (1935); 203, 147 (1936); S. N. Bernstein, Works, 2, 1954, p. 170.
\({}^{2}\) D. A. Raikov, DAN, 24, 652 (1939).
\({}^{3}\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 10, 207 (1946).
\({}^{4}\) A. F. Timan, Scientific Notes of Dnepropetrovsk Univ., 153 (1948).
\({}^{5}\) A. Zygmund, Trigonometric Series, 1939.
\({}^{6}\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, 1960.