V. V. Zhuk

On the Absolute Convergence of Fourier Series

(Presented by Academician S. N. Bernstein, July 15, 1964)

Let \(f(x)\in L^2_{2\pi}\) be a \(2\pi\)-periodic function. Put

\[ {}^{s}\Delta_t^p f(x)=\sum_{k=0}^{p}(-1)^k C_p^k f[x+(p-2k)t], \]

\[ L^{(p)}(h,x,f)=\frac{1}{h}\int_0^h {}^{s}\Delta_t^p f(x)\,dt, \]

\[ L_2^{(p)}(h,f)=\frac{1}{h}\left\{\sup_{0\le u\le h}\int_{-\pi}^{\pi}\left[\int_0^u {}^{s}\Delta_t^p f(x)\,dt\right]^2 dx\right\}^{1/2}, \]

\[ {}^{s}\omega_p(h,f)=\sup_{0\le t\le h}\ \sup_{-\pi\le x\le \pi}\left|{}^{s}\Delta_t^p f(x)\right|, \]

\[ {}^{s}\omega_p^{(2)}(h,f)=\left\{\sup_{0\le t\le h}\int_{-\pi}^{\pi}\left[{}^{s}\Delta_t^p f(x)\right]^2 dx\right\}^{1/2}. \]

Lemma 1. Suppose

\[ f(x)\sim \frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx). \]

Then

\[ \frac{1}{n^{2p}}\sum_{k=1}^{n}(a_k^2+b_k^2)k^{2p}\le C(p)\int_{-\pi}^{\pi}\left[L^{(p)}\left(\frac{1}{n},x,f\right)\right]^2 dx, \]

where \(p\) is any natural number, and \(C(p)\) is a constant depending only on \(p\).

Lemma 2. If \(0<m\le 2\), then

\[ \sum_{k=2^{\gamma-1}+1}^{2^\gamma}\left(|a_k|^m+|b_k|^m\right) \le C_1(p)\left\{\int_{-\pi}^{\pi}\left[L^{(p)}\left(\frac{1}{2^\gamma},x,f\right)\right]^2 dx\right\}^{m/2}(2^\gamma)^{1-m/2}, \]

where \(p\) and \(\gamma\) are any natural numbers, and \(C_1(p)\) is a constant depending only on \(p\).

Lemma 3. If \(2^{l-1}\le n<2^l\), then

\[ \sum_{k=n}^{\infty} k^{\alpha-1}E\left(\frac{1}{k}\right) \ge C(\alpha)\sum_{\gamma=l+1}^{\infty}(2^\gamma)^\alpha E\left(\frac{1}{2^\gamma}\right), \]

where \(E\left(\frac{1}{k}\right)\downarrow 0\) as \(k\uparrow\infty\); \(\alpha\) is any real number; \(C(\alpha)\) is a constant depending only on \(\alpha\); \(n\) and \(l\) are natural numbers.

Theorem 1. If \(0<m\leqslant 2\), then

\[ \sum_{k=n}^{\infty}\left(|a_k|^m+|b_k|^m\right)\leqslant \]

\[ \leqslant C(p,m)\left\{\sum_{k=n}^{\infty} \frac{\left[L_2^{(p)}\left(\frac1k,f\right)\right]^m}{k^{m/2}} +\left[L_2^{(p)}\left(\frac1n,f\right)\right]^m n^{1-m/2}\right\}, \]

where \(C(p,m)<+\infty\) is a constant depending only on \(p\) and \(m\).

Proof. Let \(2^{l-1}\leqslant n<2^l\). Then

\[ \sum_{k=n}^{\infty}\left(|a_k|^m+|b_k|^m\right) \leqslant \sum_{k=n}^{2^l}\left(|a_k|^m+|b_k|^m\right) +\sum_{\gamma=l+1}^{\infty}\sum_{k=2^{\gamma-1}+1}^{2^\gamma} \left(|a_k|^m+|b_k|^m\right) = \]

\[ =O\left( \sum_{k=n}^{2n}\left(|a_k|^m+|b_k|^m\right) + \sum_{\gamma=l+1}^{\infty} \left\{ \int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1{2^\gamma},x,f\right)\right]^2 dx \right\}^{m/2} (2^\gamma)^{1-m/2} \right)=I_1+I_2. \]

Let us estimate \(I_1\):

\[ \sum_{k=n}^{2n}\left(|a_k|^m+|b_k|^m\right) \leqslant \left\{\sum_{k=n}^{2n}(a_k^2+b_k^2)\right\}^{m/2} n^{1-m/2} \leqslant \]

\[ \leqslant C_2(p)\left\{ \int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1{2n},x,f\right)\right]^2 dx \right\}^{m/2} n^{1-m/2} \leqslant C_3(p)\left[L_2^{(p)}\left(\frac1n,f\right)\right]^m n^{1-m/2}. \]

Passing to the estimate of \(I_2\), we have

\[ \sum_{\gamma=l+1}^{\infty} \left\{ \int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1{2^\gamma},x,f\right)\right]^2 dx \right\}^{m/2} (2^\gamma)^{1-m/2} \leqslant \sum_{\gamma=l+1}^{\infty} \left[L_2^{(p)}\left(\frac1{2^\gamma},f\right)\right]^m (2^\gamma)^{1-m/2}. \]

Applying* Lemma 3, we obtain

\[ \sum_{\gamma=l+1}^{\infty} \left[L_2^{(p)}\left(\frac1{2^\gamma},f\right)\right]^m (2^\gamma)^{1-m/2} \leqslant C(m)\sum_{k=n}^{\infty} \frac{\left[L_2^{(p)}\left(\frac1k,f\right)\right]^m}{k^{m/2}}. \]

The rest is clear.

Corollary 1. It is not hard to verify that

\[ L_2^{(p)}(h,f)\leqslant { }^{s}\omega_p^{(2)}(h,f). \]

Consequently, for \(0<m\leqslant 2\),

\[ \sum_{k=n}^{\infty}\left(|a_k|^m+|b_k|^m\right) = O\left( \sum_{k=n}^{\infty} \frac{\left[{ }^{s}\omega_p^2\left(\frac1k,f\right)\right]^m}{k^{m/2}} \right) + \left[{ }^{s}\omega_p^{(2)}\left(\frac1n,f\right)\right]^m n^{1-m/2}. \]

In particular, if \(f(x)\) has \(l\) derivatives, \(f^{(l)}(x)\in \mathrm{Lip}\,\alpha\), and \(m(l+\alpha)+m/2>1\), then

\[ \sum_{k=n}^{\infty}\left(|a_k|^m+|b_k|^m\right) = O\left(n^{\,1-m(l+\alpha+1/2)}\right). \]

For \(l=0\) this is a result of Lorentz \((^1)\).

\[ * \ \text{For } E\left(\frac1k\right) \text{ we take } \left\{ \sup_{0\leqslant u\leqslant 1/k} \int_{-\pi}^{\pi} \left[ \int_0^u { }^{s}\Delta_t^p f(x)\,dt \right]^2 dx \right\}^{1/2}. \]

Corollary 2. If

\[ \sup_{0<t\leq 1/n}\int_{-\pi}^{\pi}\left|\,{}^{s}\Delta_t^p f(x)\right|\,dx =O\left(\frac1n\right),\qquad \sum_{k=1}^{\infty}\frac{\left[{}^{s}\omega_p\left(\frac1k,f\right)\right]^{1/2}}{k}<+\infty, \]

then

\[ \sum_{k=1}^{\infty}\bigl(|a_k|+|b_k|\bigr)<+\infty . \tag{1} \]

From this, in particular, the following two theorems follow:

Theorem A (A. Zygmund \((^2)\)). If \(f(x)\) has bounded variation and*

\[ \sum_{k=1}^{\infty} \frac{\sqrt{\omega\left(\frac1k,f\right)}}{k}<+\infty, \]

where

\[ \omega(\delta,f)=\sup_{|x_1-x_2|<\delta}|f(x_1)-f(x_2)|, \]

then relation (1) is satisfied.

Theorem B (F. I. Harshiladze \((^4)\)). If \(f(x)\) is such that for all \(x\) and \(h\)

\[ |f(x+h)+f(x-h)-2f(x)|\leq Mh^\alpha \]

with \(\alpha>0\), and, moreover, \(f(x)\) has bounded second variation on \([0,2\pi]\), then its Fourier series converges absolutely.

Indeed, Theorem A is obvious, while Theorem B follows from the equality

\[ \int_{-\pi}^{\pi}|f(x+t)-2f(x)+f(x-t)|\,dx=O(t), \tag{2} \]

which is valid whenever \(f(x)\) has bounded second variation \((^5)\). Since \((^6)\) it does not follow from (2) that \(f\in V_2\), it is clear that Corollary 2 is stronger than Theorems A and B.

Let \(F(x)\) be an antiderivative of \(f(x)\).

Theorem 1. If

\[ \sum_{k=1}^{\infty} k^{1/2}\omega_2^{(2)}\left(F,\frac1k\right)<+\infty, \]

then relation (1) is satisfied.

Corollary 1. (Theorem of Sas \((^7)\)). If

\[ \sum_{k=1}^{\infty}\frac{\omega_1^{(2)}(1/k,f)}{\sqrt{k}}<+\infty, \]

then relation (1) is satisfied.

Corollary 2. From

\[ \sum_{k=1}^{\infty} \left\{ k\,{}^{s}\omega_2\left(F,\frac1k\right) \sup_{0<t\leq 1/k} \int_{-\pi}^{\pi}|F(x+t)-2F(x)+F(x-t)|\,dx \right\}^{1/2}<+\infty \]

it follows that (1) holds.

Corollary 3. If

\[ \sum_{k=1}^{\infty}\sqrt{k}\,{}^{s}\omega_2\left(F,\frac1k\right)<+\infty, \]

then (1) holds.

* The assertion formulated in the text is due to Salem \((^3)\), who notes that it follows immediately from Zygmund’s results.

Hence it follows

Theorem (S. N. Bernstein \(^{8}\)). The relation

\[ \sum_{k=1}^{\infty} \frac{\omega(1/k,f)}{\sqrt{k}} < +\infty \]

implies (1).

Corollary 4. If

\[ \sup_{0\le t\le 1/n}\int_{-\pi}^{\pi} \left|F(x+t)-2F(x)+F(x-t)\right|\,dx =O\left(\frac{1}{n^{2}}\right), \]

then from

\[ \sum_{k=1}^{\infty}\frac{\sqrt[5]{\omega_{2}(F,1/k)}}{\sqrt{k}}<+\infty \]

follows (1).

This assertion contains the theorem of A. Zygmund mentioned above.

Leningrad State University
named after A. A. Zhdanov

Received
12 VI 1964

REFERENCES

\(^{1}\) G. G. Lorentz, Math. Zs., 51, 135 (1948).
\(^{2}\) A. Zygmund, J. London Math. Soc., 3, 194 (1928).
\(^{3}\) R. Salem, Duke Math. J., 10, 23 (1943).
\(^{4}\) F. I. Kharshiladze, DAN, 79, No. 2, 201 (1951).
\(^{5}\) F. I. Kharshiladze, Tr. Tbilissk. matem. inst., 20, 145 (1954).
\(^{6}\) F. I. Kharshiladze, Tr. Tbilissk. univ., 64, 94 (1957).
\(^{7}\) O. Szász, Ann. Math., 47, 213 (1946).
\(^{8}\) S. N. Bernstein, Collected Works, 2, 1954, pp. 166–169.