Reports of the Academy of Sciences of the USSR
1961. Vol. 141, No. 3

MATHEMATICS

M. K. POTAPOV

ON THE FOURIER COEFFICIENTS OF PERIODIC FUNCTIONS BELONGING TO THE \(H\)-CLASSES OF S. M. NIKOLSKII

(Presented by Academician A. N. Kolmogorov on 23 VI 1961)

I. As usual \((^1)\), we shall say that a function \(f(x)\), periodic with period \(2\pi\), belongs to the class \(H_p^{(r)}\), where \(1 \le p \le \infty\), \(r=\bar r+\alpha\), \(0<\alpha\le 1\), \(\bar r\) is an integer, if \(f(x)\) has an \(\bar r\)-th derivative such that \(f^{(\bar r)}(x)\in L_p\) and

\[ \bigl\|f^{(\bar r)}(x+h)-2f^{(\bar r)}(x)+f^{(\bar r)}(x-h)\bigr\|_{L_p}\le |h|^\alpha, \]

where in this inequality, for \(0<\alpha<1\), the second difference may be replaced by the first. By \(E_n(f)_{L_p}\) we shall denote the best approximation, in the metric \(L_p\), of the periodic function \(f(x)\) by trigonometric polynomials of order \(n\).

It is well known \((^1)\):

Theorem A. In order that \(f(x)\in H_p^{(r)}\), it is necessary and sufficient that \(E_n(f)_{L_p}\le C/n^r\).

Since we are dealing with periodic functions, it would naturally be desirable to pose the question of the existence of conditions, simultaneously necessary and sufficient for \(f(x)\in H_p^{(r)}\), not in terms of best approximations, but in terms of the Fourier coefficients of the function \(f(x)\). For even and odd functions such conditions are known \((^{2,3})\), namely: if

\[ f(x)\sim \sum_{k=1}^{\infty} a_k\cos kx \]

or

\[ f(x)\sim \sum_{k=1}^{\infty} a_k\sin kx \]

and \(a_k\downarrow\), then the conditions

\[ |a_k|\le C/k^{r+1-1/p} \]

are necessary and sufficient in order that \(f(x)\in H_p^{(r)}\), \(1<p\le\infty\). From Theorems 3, 4, and 5 given below it follows, in particular, that in the general case no such conditions exist.

II. Theorem 1. In order that a periodic function

\[ f(x)\sim \sum_{k=-\infty}^{\infty} C_k e^{ikx} \]

belong to the class \(H_p^{(r)}\), \(1\le p\le\infty\), it is sufficient that its Fourier coefficients satisfy the following conditions:

\[ \text{for } 2\le p\le\infty \qquad \left(\sum_{|k|=n+1}^{\infty}|C_k|^{p'}\right)^{1/p'}\le \frac{C}{n^r}, \qquad \frac{1}{p}+\frac{1}{p'}=1; \tag{1} \]

\[ \text{for } 1\le p\le 2 \qquad \left(\sum_{|k|=n+1}^{\infty}|C_k|^2\right)^{1/2}\le \frac{C}{n^r}. \tag{2} \]

Proof. Obviously, for any \(p\), \(1\le p\le\infty\),

\[ E_n(f)\le \|f(x)-s_n(x)\|_{L_p}\, *. \]

Let \(2\le p\le\infty\); then, on the basis of the theo—

\(*\) Here and below, \(s_n(x)\) denotes the partial sum of the Fourier series for \(f(x)\).

by the Hausdorff—Young theorem ((4), p. 131), it follows from this inequality that

\[ E_n(f)_{L_p}\leqslant \left(\sum_{|k|=n+1}^{\infty}|C_k|^{p'}\right)^{1/p'} . \tag{3} \]

Let \(1\leqslant p\leqslant 2\); then, obviously,

\[ E_n(f)_{L_p}\leqslant C E_n(f)_{L_2} = C\left(\sum_{|k|=n+1}^{\infty}|C_k|^2\right)^{1/2}; \tag{4} \]

if now \(f(x)\) satisfies the conditions of Theorem 1, then from inequalities (3) or (4) and Theorem A there follows the validity of Theorem 1. We note that Theorem 1 cannot be strengthened (see below the example to Theorem 3).

Theorem 2. If a periodic function

\[ f(x)\sim \sum_{k=-\infty}^{\infty} C_k e^{ikx} \]

belongs to the class \(H_p^{(r)}\), \(1<p\leqslant \infty\), then its Fourier coefficients must satisfy the following conditions:

\[ \text{for } 2\leqslant p\leqslant \infty \qquad \left(\sum_{|k|=n+1}^{\infty}|C_k|^2\right)^{1/2}\leqslant \frac{C}{n^r}; \tag{5} \]

\[ \text{for } 1<p\leqslant 2 \qquad \left(\sum_{|k|=n+1}^{\infty}|C_k|^{p'}\right)^{1/p'}\leqslant \frac{C}{n^r}, \qquad \frac1p+\frac1{p'}=1. \tag{6} \]

Proof. It is easy to show (see, for example, (5), p. 77) that for \(1<p<\infty\)

\[ \|f(x)-s_n(x)\|_{L_p}\leqslant C E_n(f)_{L_p}. \]

Therefore, if \(f(x)\in H_p^{(r)}\), then for \(1<p\leqslant 2\)

\[ \|f(x)-s_n(x)\|_{L_p}\leqslant C/n^r, \]

whence, on the basis of the Hausdorff—Young theorem, there follows the validity of inequality (6). Let now \(2\leqslant p\leqslant \infty\). Then, obviously,

\[ E_n(f)_{L_2}\leqslant C E_n(f)_{L_p}, \]

whence, on the basis of Parseval’s equality and the conditions of the theorem, there follows the validity of inequality (5). The theorem is proved. We note that Theorem 2 cannot be sharpened (see below the examples to Theorems 4 and 5).

As a consequence of Theorem 1, the following is proved quite simply.

Theorem 3. In order that a periodic function

\[ f(x)\sim \sum_{k=-\infty}^{\infty} C_k e^{ikx} \]

belong to the class \(H_p^{(r)}\), \(1\leqslant p\leqslant \infty\), it is sufficient that its Fourier coefficients satisfy the following conditions:

\[ \text{for } 1\leqslant p\leqslant 2 \qquad |C_k|\leqslant C/(|k|+1)^{r+1/2}; \tag{7} \]

\[ \text{for } 2\leqslant p\leqslant \infty \qquad |C_k|\leqslant C/(|k|+1)^{r+1-1/p}. \tag{8} \]

Let us give examples showing that Theorem 3 cannot be strengthened.

Case \(2\leqslant p\leqslant \infty\). Consider

\[ f(x)\sim \sum_{n=1}^{\infty}\frac{\cos(nx-r\pi/2)}{n^{r+1-1/p}} . \]

The function \(f(x)\) satisfies the conditions of Theorem 3; therefore it belongs to the class \(H_p^{(r)}\). We shall show that \(f(x)\) cannot have a derivative in the sense of Weyl of order \(r\) belonging to \(L_p\), i.e. \(f(x)\notin W_p^{(r)}\). Hence it will follow that \(f(x)\) cannot belong to the better class \(H_p^{(\rho)}\), where \(\rho>r\), since \(H_p^{(\rho)}\subset W_p^{(r)}\subset H_p^{(r)}\).

Suppose the contrary—that \(f(x)\) has a derivative of order \(r\) belonging to \(L_p\), i.e. \(f^{(r)}(x)\in L_p\); then we would have

\[ f^{(r)}(x)\sim \]

\[ \sim \sum_{k=1}^{\infty} a_k \cos kx,\quad \text{where}\quad a_k=\frac{1}{k^{1-1/p}}. \]
By assumption, \(f^{(r)}(x)\) must belong to \(L_p\). But in order that, for \(1<p<\infty\), the function
\[ \varphi(x)\sim \sum_{k=1}^{\infty} a_k\cos kx,\quad \text{where } a_1\geq a_2\geq\cdots\to 0, \]
belong to \(L_p\), it is necessary and sufficient ((4), p. 212) that the expression
\[ \sum_{k=1}^{\infty} a_k^p k^{p-2} \]
be finite; in our case it is infinite. Hence, for \(2\leq p<\infty\) our assumption is false and the theorem cannot be strengthened.

For \(p=\infty\), the function \(f^{(r)}(x)=-\log 2\sin \tfrac12 x\), and this function does not belong to \(L_\infty\); hence, for \(p=\infty\) too, our assumption is false and the theorem cannot be strengthened.

Case \(1\leq p\leq 2\). Consider
\[ f(x)\sim \sum_{k=1}^{\infty}\frac{\varepsilon_k}{k^{r+1/2}}\cos\left(nx-r\frac{\pi}{2}\right), \]
where \(\varepsilon_k=\pm 1\) and the plus and minus signs are chosen so that the series
\[ \sum_{k=1}^{\infty}\pm \frac{\cos kx}{k^{1/2}} \]
is not a Fourier series. The function \(f(x)\) satisfies the conditions of Theorem 3, and therefore it belongs to the class \(H_p^{(r)}\). To show that Theorem 3 cannot be strengthened, it suffices, as above, to show that the function \(f(x)\) cannot have a derivative of order \(r\) belonging to \(L_p\). Assuming the contrary, we would have \(f^{(r)}(x)\in L_p\) and
\[ f^{(r)}(x)\sim \sum_{k=1}^{\infty}\pm \frac{\cos kx}{k^{1/2}} . \tag{9} \]
Since the series (9) is not a Fourier series, our assumption about the existence of \(f^{(r)}(x)\in L_p\) is false, and this means that the theorem cannot be strengthened.

Theorem 4. If the periodic function
\[ f(x)\sim \sum_{k=-\infty}^{\infty} C_k e^{ikx} \]
belongs to the class \(H_p^{(r)}\), then its Fourier coefficients necessarily satisfy the conditions
\[ |C_k|\leq C/(|k|+1)^r,\qquad |k|=0,1,2,\ldots . \tag{10} \]

Theorem 4 is known (see, for example, (4), p. 22); it is included here for completeness of exposition. Theorem 4 cannot be sharpened. Indeed, the function
\[ f(x)=\sum_{k=1}^{\infty}\frac{1}{2^{kr}}\cos\left(2^kx+r\frac{\pi}{2}\right) \]
satisfies the conditions of Theorem 4, i.e. \(|C_k|\leq C/(|k|+1)^r\) and \(f(x)\in H_p^{(r)}\), but \(f(x)\notin W_p^{(r)}\).

Remark. Assuming that \(|C_k|\downarrow\), Theorem 4 can be sharpened (see Theorem 5). However, as the examples to Theorem 3 show, the assumption that \(|C_k|\downarrow\) does not allow Theorem 3 to be sharpened.

Theorem 5. If the periodic function
\[ f(x)\sim \sum_{k=-\infty}^{\infty} C_k e^{ikx} \]
belongs to the class \(H_p^{(r)}\), \(1\leq p\leq \infty\), and
\[ 2|C_0|\geq |C_1|+|C_{-1}|\geq |C_2|+|C_{-2}|\geq \cdots, \]
i.e. \(|C_k|\downarrow\), then its Fourier coefficients necessarily satisfy the following conditions:
\[ \text{for } 2\leq p<\infty \qquad |C_k|\leq C/(|k|+1)^{r+1/2}; \tag{11} \]
\[ \text{for } 1\leq p\leq 2 \qquad |C_k|\leq C/(|k|+1)^{r+1-1/p}. \tag{12} \]

Proof. Let \(|C_k|\downarrow\), i.e.
\[ 2|C_0|\ge |C_1|+|C_{-1}|\ge |C_2|+|C_{-2}|\ge\cdots; \]
then
\[ \sum_{k=[n/2]+1}^{n}\frac{|C_k|}{|k|} \ge (|C_n|+|C_{-n}|)\left(\frac{1}{[n/2]+1}+\cdots+\frac{1}{n}\right) \ge \frac{|C_n|+|C_{-n}|}{2}, \]
whence
\[ |C_n|+|C_{-n}|\le \]
\[ \le 2\sum_{k=[n/2]+1}^{\infty}\frac{|C_k|}{|k|} \le 2\left(\sum_{|k|=[n/2]+1}^{\infty}\frac{1}{|k|^p}\right)^{1/p} \left(\sum_{|k|=[n/2]+1}^{\infty}|C_k|^{p'}\right)^{1/p'}, \tag{13} \]
\[ \frac{1}{p}+\frac{1}{p'}=1. \]

Let the function \(f(x)\) satisfy the conditions of Theorem 5. Then for \(1<p\le 2\) the validity of Theorem 5 follows from inequality (13) and Theorem 2, while for \(2\le p\le\infty\) the validity of Theorem 5 follows from inequality (13), if in it one puts \(p=p'=2\), and also from Theorem 2. For \(p=1\), Theorem 5 is a particular case of Theorem 4.

We give examples showing that Theorem 5 cannot be sharpened.

Case \(2\le p\le\infty\). Consider
\[ f(x)=\sum_{n=2}^{\infty}\frac{e^{i n\ln n}}{n^{r+1/2}}e^{inx}. \]
This function has Fourier coefficients satisfying inequality (11) and such that \(|C_k|\downarrow\). We shall show that \(f(x)\in H_p^{(r)}\) for every \(p\).

Indeed, for \(p=\infty\) the function \(f(x)\in H_\infty^{(r)}\), for if \(r=\bar r+\alpha\), where \(0<\alpha\le 1\), and \(\bar r\) is an integer, then
\[ f^{(\bar r)}(x)=\sum_{n=2}^{\infty}\frac{e^{i n\ln n}}{n^{\alpha+1/2}}e^{inx}\in H_\infty^{(\alpha)} \]
(see \((^4)\), p. 140)*, i.e. \(f(x)\in H_p^{(r)}\) for every \(p\). We further show that \(f(x)\notin W_p^{(r)}\). Assuming the contrary, we obtain that \(f^{(r)}(x)\in L_p\) and
\[ f^{(r)}(x)\sim \sum_{n=2}^{\infty}\frac{e^{i n\ln n}}{n^{1/2}}e^{inx}. \]
However, this is not so, since for \(p=\infty\), \(f^{(r)}(x)\notin L_\infty\) (see \((^4)\), p. 122), and for \(p=2\), \(\|f^{(r)}(x)\|_{L_2}=\infty\), i.e. \(f^{(r)}(x)\notin L_2\). This means that \(f^{(r)}(x)\notin L_p\) for all \(p\) in the interval \(2\le p\le\infty\), i.e. Theorem 5 in this case is not sharpenable.

Case \(1<p\le 2\). Consider
\[ f(x)\sim \sum_{k=1}^{\infty}\frac{\cos(kx-r\pi/2)}{k^{r+1-1/p}}. \]
This function has Fourier coefficients satisfying inequality (12) and such that \(|C_k|\downarrow\). Moreover, it belongs to the class \(H_p^{(r)}\), \(1<p\le 2\) (see \((^2)\), p. 436), and does not belong to a better class \(f(x)\notin W_p^{(r)}\) (see the example to Theorem 3), i.e. Theorem 5 is not sharpened in this case either.

The case \(p=1\) has already been considered earlier (see the example to Theorem 4).

Moscow State University
named after M. V. Lomonosov

Received
23 VI 1961

References

1. S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 44 (1951).
2. A. A. Konjushkov, Izv. AN SSSR, ser. matem., 21, No. 3, 423 (1957).
3. G. G. Lorentz, Math. Zs., 51, No. 2 (1948).
4. A. Zygmund, Trigonometric Series, Moscow–Leningrad, 1933.
5. M. K. Potapov, Uch. zap. Ivanovsk. gos. ped. inst., 18 (1958).

* In \((^4)\) it is proved that \(f^{(\bar r)}(x)\in H_\infty^\alpha\) only for \(0<\alpha<1\); however, for \(\alpha=1\) the proof is analogous.