UDC 517

MATHEMATICS

M. F. TIMAN

ON THE ORDER OF APPROXIMATION OF A FUNCTION BY ZYGMUND NORMAL MEANS

(Presented by Academician L. V. Kantorovich on 23 X 1967)

Let \(L_p\) \((1 \leqslant p \leqslant \infty)\) denote the space of measurable \(2\pi\)-periodic functions for which
\[ \|f(x)\|_{L_p}=\left\{\int_0^{2\pi}|f(x)|^p\,dx\right\}^{1/p}<\infty \quad (1\leqslant p<\infty), \]
and, for \(p=\infty\),
\[ \operatorname*{vrai\,sup}_x |f(x)|<\infty. \]
For each function \(f(x)\in L_p\) with Fourier series
\[ \sum_{\nu=0}^{\infty} A_\nu(x), \quad \left(A_0(x)\equiv \frac{a_0}{2},\quad A_\nu(x)=a_\nu\cos \nu x+b_\nu\sin \nu x,\quad \nu=1,2,\ldots\right), \]
for any natural \(k\) we consider the operator
\[ Z_n^{(k)}(f;x)= \sum_{\nu=0}^{n}\left(1-\frac{\nu^k}{(n+1)^k}\right)A_\nu(x), \]
which is usually called the Zygmund normal mean of order \(k\).

Numerous works have been devoted to the investigation of the order of approximation of each individual function \(f(x)\in L_p\) by the operators \(Z_n^{(k)}(f;x)\), depending on its structural properties. At the same time, as Zygmund already showed in \(\left({}^{1}\right)\), there is a substantial difference in considering this question for the cases of even and odd \(k\).

If \(k\) is an even number, and the function \(f(x)\) has a \((k-1)\)-st absolutely continuous derivative and \(\|f^{(k)}(x)\|_{L_\infty}<\infty\), then (see \(\left({}^{1}\right)\))
\[ \|f(x)-Z_n^{(k)}(f;x)\|_{L_\infty} \leqslant c_k(n+1)^{-k}\|f^{(k)}(x)\|_{L_\infty}. \tag{1} \]
Another proof of this Zygmund inequality was given by the author in \(\left({}^{8}\right)\). It is based on the obvious estimate
\[ \|f(x)-Z_n^{(k)}(f;x)\|_{L_p} \leqslant (1+\|Z_n^{(k)}\|)\|f(x)-T_n(x)\|_{L_p} +(n+1)^{-k}\left\|\frac{d^k}{dx^k}T_n(x)\right\|_{L_p} \tag{2} \]
valid for any trigonometric polynomial \(T_n(x)\) (see \(\left({}^{8}\right)\) for \(p=\infty\) and \(\left({}^{7}\right)\), p. 591, for any \(1\leqslant p\leqslant\infty\)).

From inequality (1) and one result of S. B. Stechkin \(\left({}^{4}\right)\) it follows that, for any continuous function \(f(x)\),
\[ \|f(x)-Z_n^{(k)}(f;x)\|_{L_\infty} \leqslant c_k\omega_k\left(f;\frac{1}{n+1}\right)_{L_\infty}, \tag{3} \]
where
\[ \omega_k(f;h)_{L_p} = \sup_{|t|\leq h}\|\Delta_t^k f(x)\|_{L_p} = \sup_{|t|\leq h} \left\| \sum_{\nu=0}^{k}(-1)^{k-\nu}\binom{k}{\nu}f(x+\nu t) \right\|_{L_p}. \]
Estimate (3) and the analogous estimate for the case where \(f(x)\in L_p\) \((1\leqslant p\leqslant\infty)\),
\[ \|f(x)-Z_n^{(k)}(f;x)\|_{L_p} \leqslant c_k\omega_k\left(f;\frac{1}{n+1}\right)_{L_p}, \tag{4} \]
follow from (2), if one uses the following inequality for pro-

derivatives of a trigonometric polynomial (see \((^{5,6})\)):

\[ \left\|\frac{d^k}{dx^k}T_n(x)\right\|_{L_p} \ll \left(\frac{n}{2}\right)^k \left\|\Delta_{\pi/2n}^{k}T_n(x)\right\|_{L_p}. \tag{5} \]

R. M. Trigub \((^{10})\) and V. V. Zhuk \((^{11})\) independently showed that, for even \(k\), for any function \(f(x)\in L_p\) the converse to inequality (4) is also true, i.e.,

\[ \omega_k\left(f;\frac{1}{n+1}\right)_{L_p} \ll c_k\left\|f(x)-Z_n^{(k)}(f;y)\right\|_{L_p} \qquad (1\le p\le\infty). \tag{6} \]

Thus, for any function \(f(x)\in L_p\) \((1\le p\le\infty)\), for even \(k\) the order relation * holds:

\[ \left\|f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} \asymp \omega_k\left(f;\frac{1}{n+1}\right)_{L_p}. \tag{7} \]

In the case when \(k\) is an odd number and \(1<p<\infty\), relation (7) also remains valid. Estimate (4) in this case for \(k=1\) was obtained by P. L. Ul'yanov \((^{9})\), and for other values of \(k\) by Yu. A. Ponomarenko \((^{12})\). Inequality (6) can be obtained with the aid of the identity

\[ \frac{d^k}{dx^k}Z_n^{(k)}(f;x) = (-1)^{k/2}(n+1)^k \left\{\widetilde Z_n^{(k)}(f;x)-\widetilde Z_{n,1}^{(k)}(f;x)\right\}, \tag{8} \]

where \(\widetilde Z_n^{(k)}(f;x)\) are the Zygmund normal means for the series conjugate to the Fourier series of the function \(f(x)\), and
\(\widetilde Z_{n,1}^{(k)}(f;x)=Z_n^{(k)}(\widetilde Z_n^{(k)}(f);x)\). Indeed, for any \(h\) \(\left(0<h\le\frac{1}{n+1}\right)\),

\[ \left\|\Delta_h^k f(x)\right\|_{L_p} \le \left\|\Delta_h^k f(x)-\Delta_h^k Z_n^{(k)}(f;x)\right\|_{L_p} + h^k \left\|\frac{d^k}{dx^k}Z_n^{(k)}(f;x)\right\|_{L_p} \le \]

\[ \le 2^k\left\|f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} + \left\|\widetilde Z_n^{(k)}(f;x)-\widetilde Z_{n,1}^{(k)}(f;x)\right\|_{L_p}. \]

Since \(1<p<\infty\), the function \(\widetilde f(x)\), conjugate to \(f(x)\), also belongs to the space \(L_p\), and, by the well-known inequality of M. Riesz (see \((^{13})\), p. 149), we find

\[ \left\|\Delta_h^k f(x)\right\|_{L_p} \le 2^k\left\|f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} + \left\|\widetilde f(x)-\widetilde Z_n^{(k)}(f;x)\right\|_{L_p} + \]

\[ + \left\|\widetilde f(x)-\widetilde Z_{n,1}^{(k)}(f;x)\right\|_{L_p} \le c_{p,k} \left\{ \left\|f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} + \left\|f(x)-Z_{n,1}^{(k)}(f;x)\right\|_{L_p} \right\}, \]

where

\[ Z_{n,1}^{(k)}(f;x) = Z_n^{(k)}(Z_n^{(k)}(f);x) = \sum_{\nu=0}^{n} \left(1-\frac{\nu^k}{(n+1)^k}\right)^2 A_\nu(x). \]

From this it is easy to obtain

\[ \left\|\Delta_h^k f(x)\right\|_{L_p} \ll M_{p,k}\left\|f(x)-Z_n^{(k)}(f;x)\right\|_{L_p}. \]

From this estimate we obtain (6), with a constant depending not only on \(k\), but also on \(p\).

For \(p=\infty\) or \(p=1\) and odd \(k\), the order relation (7), as is known, does not hold. In these cases, when studying the order of approximation of a function \(f(x)\) by the operators \(Z_n^{(k)}(f;x)\), as has been shown in a number of works (see, for example, \((^{1-4})\)), it is important to take into account the structural properties not only of the function \(f(x)\) itself, but also the properties of the function \(\widetilde f(x)\) conjugate to it.

Below the following assertion is established concerning the approximation of a function \(f(x)\in L_p\) \((1\le p\le\infty)\) by the Zygmund normal means \(Z_n^{(k)}(f;x)\) for the case when \(k\) is any odd number.

\[ \text{* The relation } u\asymp v \text{ means that } c_1v\le u\le c_2v,\text{ where } c_1>0,\ c_2>0 \text{ are some constants.} \]

Theorem. If \(f(x)\in L_p\) \((1\le p\le \infty)\) has Fourier series \(\sum_{\nu=0}^{\infty} A_\nu(x)\), then for any odd \(k\) the following order relation holds:

\[ \left\| f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} \preccurlyeq \omega_{k+1}\left(f;\frac{1}{n+1}\right)_{L_p} + \omega_{k+1}\left(\widetilde F;\frac{1}{n+1}\right)_{L_p}(n+1), \tag{9} \]

where \(\widetilde F(x)\) is a function having Fourier series \(\sum_{\nu=1}^{\infty}\frac{1}{\nu}A_\nu(x)\).*

The proof of the theorem is based on the following, easily verified identities:**

\[ (n+1)^{-k-1}\frac{d^{k+1}}{dx^{k+1}}Z_n^{(k)}(f;x) = (-1)^{(k-1)/2} \left[ Z_n^{(k)}(f;x)-Z_{n,1}^{(k)}(f;x)-Z_n^{(k)}(Z_n^{(1)}(f); \right. \]

\[ \left. x)+Z_{n,1}^{(k)}(Z_n^{(1)}(f);x) \right], \tag{10} \]

\[ (n+1)^{-k}\frac{d^{k+1}}{dx^{k+1}}Z_n^{(k+1)}(\widetilde F;x) = (-1)^{(k-1)/2} \left[ Z_n^{(k+1)}(f;x)-Z_n^{(k+1)}(Z_n^{(k)}(f);x) \right], \tag{11} \]

\[ Z_n^{(k+1)}(f;x)=Z_n^{(k)}(f;x)+Z_n^{(1)}(f;x)-Z_n^{(1)}(Z_n^{(k)}(f);x). \tag{12} \]

Let us first establish that for any \(p\) \((1\le p\le \infty)\) and odd \(k\)

\[ \left\| f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} \preccurlyeq c_k\left\{ \omega_{k+1}\left(f;\frac{1}{n+1}\right)_{L_p} + (n+1)\omega_{k+1}\left(\widetilde F;\frac{1}{n+1}\right)_{L_p} \right\}. \tag{13} \]

Obviously,

\[ \begin{aligned} \left\| f(x)-Z_n^{(k)}(f;x)\right\|_{L_p} &\le \left\| f(x)-Z_n^{(k+1)}(f;x)\right\|_{L_p} + \left\| Z_n^{(k)}(f-Z_n^{(k+1)}(f);x)\right\|_{L_p} \\ &\quad+ \left\| Z_n^{(k+1)}(f;x)-Z_n^{(k)}(Z_n^{(k+1)}(f);x)\right\|_{L_p} = U_1+U_2+U_3. \end{aligned} \tag{14} \]

Since \(k+1\) is an even number, by (4) and the boundedness of the norm of the operator \(Z_n^{(k)}(f;x)\), we find

\[ U_1\preccurlyeq c_k\omega_{k+1}\left(f;\frac{1}{n+1}\right)_{L_p}, \qquad U_2\preccurlyeq b_k\omega_{k+1}\left(f;\frac{1}{n+1}\right)_{L_p}. \tag{15} \]

Using identity (11) and inequalities (5) and (4), we obtain

\[ \begin{aligned} U_3 &\preccurlyeq (n+1)^{-k} \left\| \frac{d^{k+1}}{dx^{k+1}}Z_n^{(k+1)}(\widetilde F;x) \right\|_{L_p} \\ &\preccurlyeq B_k(n+1)\omega_{k+1}\left(Z_n^{(k+1)}(\widetilde F);\frac{1}{n+1}\right)_{L_p} \\ &\preccurlyeq M_k(n+1) \left\{ \omega_{k+1}\left(\widetilde F;\frac{1}{n+1}\right)_{L_p} + \left\|\widetilde F(x)-Z_n^{(k+1)}(\widetilde F;x)\right\|_{L_p} \right\} \\ &\preccurlyeq M_k'(n+1)\omega_{k+1}\left(\widetilde F;\frac{1}{n+1}\right)_{L_p}. \end{aligned} \tag{16} \]

From estimates (15), (16), and (14), inequality (13) follows.

We shall now show that for any odd \(k\) the inequalities

\[ \omega_{k+1}\left(f;\frac{1}{n+1}\right)_{L_p} \preccurlyeq c_k\left\| f(x)-Z_n^{(k)}(f;x)\right\|_{L_p}, \tag{17} \]

\[ \omega_{k+1}\left(\widetilde F;\frac{1}{n+1}\right)_{L_p} \preccurlyeq \frac{c_k}{n+1} \left\| f(x)-Z_n^{(k)}(f;x)\right\|_{L_p}. \tag{18} \]

* We note that in the special case when \(k=1\) (Fejér sum), relation (9) was announced, without indication of its proof, by V. V. Zhuk at the interuniversity summer scientific school on summability theory, Sverdlovsk, July 1967.

** Such identities, with an indication of their application to estimates of deviations of functions from their normal Zygmund means, were earlier presented by the author in a report at the interuniversity seminar on function theory, Dnepropetrovsk, April 21, 1967, and at the summer scientific school, Sverdlovsk, July 1967.

Let \(0<h\le \dfrac{1}{n+1}\). Using identity (10), we find

\[ \begin{aligned} \|\Delta_h^{k+1} f(x)\|_{L_p} &\le 2^{k+1}\|f(x)-Z_n^k(f;x)\|_{L_p} +\|\Delta_h^{k+1} Z_n^{(k)}(f;x)\|_{L_p} \\ &\le 2^{k+1}\|f(x)-Z_n^{(k)}(f;x)\|_{L_p} +h^{k+1}\left\|\frac{d^{k+1}}{dx^{k+1}}Z_n^{(k)}(f;x)\right\|_{L_p} \\ &\le 2^{k+1}\|f(x)-Z_n^{(k)}(f;x)\|_{L_p} \\ &\quad+\|Z_n^{(k)}(f;x)-Z_{n,1}^{(k)}(f;x) -Z_n^k(Z_n^{(1)}(f);x)+Z_{n,1}^{(k)}(Z_n^{(1)}(f);x)\|_{L_p}. \end{aligned} \tag{19} \]

Adding and subtracting under the norm sign the difference \(f(x)-Z_n^{(1)}(f;x)\), we obtain

\[ \begin{aligned} &\|Z_n^{(k)}(f;x)-Z_{n,1}^{(k)}(f;x) -Z_n^{(k)}(Z_n^{(1)}(f);x)+Z_{n,1}^{(k)}(Z_n^{(1)}(f);x)\|_{L_p} \\ &\le \|f(x)-Z_n^{(k)}(f;x)\|_{L_p} +\|f(x)-Z_{n,1}^{(k)}(f;x)\|_{L_p} \\ &\quad+\|Z_n^{(k)}(f-Z_n^{(k)}(f);x)\|_{L_p} +\|Z_n^{(1)}(f-Z_n^{(k)}(f);x)\|_{L_p}. \end{aligned} \tag{20} \]

Since, obviously,

\[ \|f(x)-Z_{n,1}^{(k)}(f;x)\|_{L_p} \le B_k\|f(x)-Z_n^{(k)}(f;x)\|_{L_p}, \]

it follows from (19) and (20) that we arrive at the estimate

\[ \|\Delta_h^{k+1} f(x)\|_{L_p} \le c_k\|f(x)-Z_n^{(k)}(f;x)\|_{L_p}. \]

In view of the fact that this inequality is valid for every \(0<h\le\dfrac{1}{n+1}\), we obtain estimate (17). Similarly, using identity (11) and identity (12) as applied to the function \(\widetilde F(x)\), for every \(h\left(0<h\le\dfrac{1}{n+1}\right)\),

\[ \begin{aligned} \|\Delta_h^{k+1}\widetilde F(x)\|_{L_p} &\le 2^{k+1}\|\widetilde F(x)-Z_n^{(k+1)}(\widetilde F;x)\|_{L_p} +h^{k+1}\left\|\frac{d^{k+1}}{dx^{k+1}}Z_n^{(k+1)}(\widetilde F;x)\right\|_{L_p} \\ &\le 2^{k+1}\|\widetilde F(x)-Z_n^{(k)}(\widetilde F;x) +Z_n^{(1)}(\widetilde F;x)-Z_n^{(1)}(Z_n^{(k)}(\widetilde F);x)\|_{L_p} \\ &\quad+\frac{1}{n+1}\|Z_n^{(k+1)}(f;x)-Z_n^{(k+1)}(Z_n^{(k)}(f);x)\|_{L_p} \\ &\le 2^{k+1}\|\varphi_k(x)-Z_n^{(1)}(\varphi_k;x)\|_{L_p} +\frac{1}{n+1}\|Z_n^{(k+1)}(f-Z_n^{(k)}(f);x)\|_{L_p}, \end{aligned} \]

where \(\varphi_k(x)=\widetilde F(x)-Z_n^{(k)}(\widetilde F;x)\). Applying to the function \(\varphi_k(x)\) the known theorem of Alexits (3), we obtain that

\[ \|\varphi_k(x)-Z_n^{(1)}(\varphi_k;x)\|_{L_p} \le C\frac{\|\widetilde{\varphi}_k(x)\|_{L_p}}{n+1} = C\frac{\|f(x)-Z_n^{(k)}(f;x)\|_{L_p}}{n+1}, \]

where \(\widetilde{\varphi}_k(x)\) is the function conjugate to \(\varphi_k(x)\). Thus,

\[ \|\Delta_h^{k+1}\widetilde F(x)\|_{L_p} \le \frac{c_k}{n+1}\|f(x)-Z_n^{(k)}(f;x)\|_{L_p}. \]

From this inequality (18) follows. From the estimates (17), (18), and (13) we arrive at the order relation (9).

Dnepropetrovsk Agricultural Institute

Received
10 X 1967

CITED LITERATURE

1. A. Zygmund, Duke Math. J., 12, 695 (1945).
2. M. Zamansky, Ann. de l’École Norm. Sup., 66, 3, 19 (1949).
3. G. Alexits, Acta Math. Hungar., 3, 29 (1952).
4. S. B. Stechkin, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 62, 48 (1961).
5. S. B. Stechkin, DAN, 60, No. 9, 1511 (1948).
6. S. M. Nikol’skii, DAN, 60, No. 9, 1507 (1948).
7. M. F. Timan, Izv. AN SSSR, ser. matem., 29, no. 3, 587 (1965).
8. V. G. Ponomarenko, M. F. Timan, Uch. zap. Kazansk. gos. univ. im. V. I. Ul’yanova (Lenina), 124, No. 6, 266 (1964).
9. P. L. Ul’yanov, Sibirsk. matem. zhurn., 5, No. 2 (1964).
10. R. M. Trigub, Izv. AN SSSR, ser. matem., 29, 615 (1965).
11. V. V. Zhuk, Studies on Certain Problems of the Constructive Theory of Functions, Leningrad Mechanical Institute, 1965, p. 93.
12. Yu. A. Ponomarenko, Some Questions of Summation of Multiple Fourier Series, Abstract of Candidate’s Dissertation, Tartu, 1965.
13. A. Zygmund, Trigonometric Series, Moscow–Leningrad, 1939.