V. V. Zhuk

ON A MODIFICATION OF THE CONCEPT OF THE MODULUS OF SMOOTHNESS AND ITS APPLICATION TO ESTIMATING FOURIER COEFFICIENTS

(Presented by Academician S. N. Bernstein on 15 VII 1964)

Let \(f(x)\in L_{2\pi}\). Put

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

Definition. Put\(^*\)

\[ L^{(p)}(h,x,f)=\frac1h\int_0^h {}^{s}\Delta_t^p f(x)\,dt. \]

The quantity

\[ L^{(p)}(h,f)=\sup_{-\pi\le x\le \pi}\left|L^{(p)}(h,x,f)\right| \]

will be called the \(L\)-modulus of smoothness of order \(p\) of the function \(f\).

Theorem 1. Let \(f(x)\in \mathscr L_{2\pi}^{\,2}\),

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

Then

\[ \frac{1}{n^{2p}}\sum_{k=1}^{n}(a_k^2+b_k^2)k^{2p} \le C(p)\int_{-\pi}^{\pi}\left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx, \]

where \(p\) is an arbitrary natural number, and \(C(p)\) depends only on \(p\).

Proof. It is easy to see that for even \(p\)

\[ {}^{s}\Delta_t^p f(x)\sim (-1)^{p/2}2^p \sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx)\sin^p kt, \]

\[ n\int_0^{1/n}{}^{s}\Delta_t^p f(x)\,dt \sim (-1)^{p/2}2^p \sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx)\, n\int_0^{1/n}\sin^p kt\,dt, \]

and for odd \(p\)

\[ n\int_0^{1/n}{}^{s}\Delta_t^p f(x)\,dt \sim (-1)^{(p-1)/2}2^p \sum_{k=1}^{\infty}(b_k\cos kx-a_k\sin kx)\, n\int_0^{1/n}\sin^p kt\,dt. \]

By Parseval’s equality,

\[ \frac1\pi\int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx = C_1(p)\sum_{k=1}^{\infty}(a_k^2+b_k^2) \left(n\int_0^{1/n}\sin kt\,dt\right)^2, \]

whence

\[ \frac{1}{n^{2p}}\sum_{k=1}^{n} k^{2p}(a_k^2+b_k^2) \le C(p)\int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx. \]

The theorem is proved.

Corollary 1. If \(f(x)\in \mathscr L_{2\pi}^{\,2}\), then

\[ |a_n|= O\left(\left\{\int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx\right\}^{1/2}\right). \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxx}} \]

\(^*\) Relying on well-known results \(\bigl(({}^1),\) pp. 83–84\(\bigr)\), it is not difficult to show that \(L^{(p)}(h,x,f)\) is a function summable in \(x\) on \([0,2\pi]\) for \(f(x)\in \mathscr L_{2\pi}\) and belongs to \(\mathscr L_{2\pi}^{\,2}\) for \(f(x)\in \mathscr L_{2\pi}^{\,2}\).

In particular,

\[ |a_n|=O\left(\left\{L^{(1)}\left(\frac1n,f\right)\right\}\right). \tag{1} \]

From this we obtain the well-known estimate

\[ |a_n|=O\left(\omega\left(\frac1n,f\right)\right). \tag{2} \]

The same estimates are also valid for \(b_n\).

Example.

\[ f(x)=\sum_{k=1}^{\infty}\frac{\cos 2^k x}{k^{1+\alpha}},\qquad 0<\alpha<1, \]

which belongs to A. Zygmund \(\bigl((^1),\) pp. 294—296\(\bigr)\), shows that

\[ L^{(1)}\left(\frac1n,f\right) = O\left(\frac1{|\ln n|^{1+\alpha}}\right) = O\left(\frac1{\ln n}\right), \]

whereas

\[ \omega\left(\frac1n,f\right)\ne O\left(\frac1{\ln n}\right). \]

Thus inequality (1) is sharper than (2).

Corollary 2. If

\[ \int_{-\pi}^{\pi}\left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx = O\left(\frac1{n^{2p}}\right), \]

then

\[ a_k=O\left(\frac1{k^p}\right),\qquad b_k=O\left(\frac1{k^p}\right). \]

Corollary 3. If \(|a_{k+1}|\le |a_k|\), then

\[ |a_n| = O\left( \frac{ \left\{\displaystyle\int_{-\pi}^{\pi} \left[L^{(p)}\left(\frac1n,x,f\right)\right]^2 dx\right\}^{1/2} }{ \sqrt n } \right). \]

Analogous assertions are also valid for \(b_n\).

Theorem 2. In order that a function \(f(x)\) whose Fourier series is lacunary have a derivative of order \(p\) belonging to \(\operatorname{Lip}\alpha\) \((0<\alpha<1)\) or to \(Z\) \((\alpha=1)\), it is necessary and sufficient that its Fourier coefficients have order \(1/n^{p+\alpha}\).

Proof. It is well known that the necessity of the condition holds even when the Fourier series is not lacunary. For a lacunary series satisfying the conditions of the theorem,

\[ \sum_{k=n}^{\infty} (|a_k|+|b_k|) = O\left(\frac1{n^{p+\alpha}}\right). \]

All the more,

\[ E_n=O\left(\frac1{n^{p+\alpha}}\right). \]

Hence, taking into account the theorems of S. N. Bernstein and A. Zygmund \(\bigl((^2),\) pp. 138—139, 145\(\bigr)\), we obtain the sufficiency of the condition.

Remark. For \(p=0\) \((0<\alpha<1)\) this theorem is known \(\bigl((^1),\) p. 691\(\bigr)\).

Corollary. It is known that if \(f(x)\) has bounded variation, then

\[ |a_k|=O\left(\frac1k\right),\qquad |b_k|=O\left(\frac1k\right). \]

It is shown by an example \(\bigl((^1),\) p. 203\(\bigr)\) that the order of this estimate cannot be improved, even if \(f(x)\) is assumed continuous. Moreover, we shall show that the indicated order cannot be improved even if the function of bounded variation \(f(x)\in Z\).

Example. Let

\[ f(x)=\sum_{m=1}^{\infty}\frac{1}{2^m}\cos 2^m x . \]

Lorentz \((^3)\) proved that if

\[ \sum_{k=n}^{\infty} (|a_k|+|b_k|)=O\left(\frac1n\right), \]

then \(f(x)\) has bounded variation. Applying Lorentz’s theorem and Theorem 2, it is easy to see that \(f(x)\) has bounded variation and belongs to \(Z\), whereas its coefficients are not \(o(1/k)\).

Theorem 3. Let \(f(x)\in L_{2\pi}^{2}\) and

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

where* \(a_k\geqslant 0\). Then, for even \(p\),

\[ \frac{1}{n^p}\sum_{k=1}^{n} a_k k^p \leqslant C(p)\left|L^{(p)}\left(\frac1n,0,f\right)\right|, \]

where \(C(p)\) is a constant depending only on \(p\).

Proof. For even \(p\) we have

\[ {}^{s}\Delta_t^p f(0)=(-1)^{p/2}2^p\sum_{k=1}^{\infty} a_k\sin^p kt, \]

\[ \left|n\int_{0}^{1/n}{}^{s}\Delta_t^p f(0)\,dt\right| \geqslant 2^p\sum_{k=1}^{n} a_k n\int_{0}^{1/n}\sin^p kt\,dt, \]

\[ \frac{1}{n^p}\sum_{k=1}^{n} a_k k^p \leqslant C(p)\left|L^{(p)}\left(\frac1n,0,f\right)\right|. \]

Corollary. If \(a_k\downarrow 0\), then

\[ a_n=O\left(\frac{L^{(p)}\left(\frac1n,0,f\right)}{n}\right). \]

In particular, if \(f(x)\) has \(m\) derivatives, and \(f^m(x)\) belongs to \(\operatorname{Lip}\alpha\) \((0<\alpha<1)\) or \(Z\) \((\alpha=1)\), then

\[ a_n=O\left(\frac{1}{n^{1+m+\alpha}}\right). \]

For functions of the form

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

with \(m=0,\ 0<\alpha<1\), this fact was established by Lorentz \((^3)\).

Thus, if \(f(x)\) has \(m\) derivatives and \(f^{(m)}(x)\in \operatorname{Lip}1\), then \(a_n=O(1/n^{2+m})\).

The example of the function

\[ \Psi(x)=\sum_{k=1}^{\infty}\frac{\cos kx}{k^2} \]

shows that this estimate cannot be improved even for \(m=0\).

* No restrictions are imposed on \(b_k\).

Theorem 4. Let

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

where \(b_k \geqslant 0\). Then, for odd \(p\), we have

\[ \frac{1}{n^p}\sum_{k=1}^{n} k^p b_k \leqslant C(p)\left|\,n^2\int_{0}^{1/n}\left[\int_{0}^{u}{}^{s}\Delta_t^p f(0)\,dt\right]du\,\right|. \]

Proof. For odd \(p\),

\[ {}^{s}\Delta_t^p f(0)=(-1)^{(p-1)/2}2^p\sum_{k=1}^{\infty} b_k\sin^p kt, \]

\[ \int_{0}^{u}{}^{s}\Delta_t^p f(0)\,dt = (-1)^{(p-1)/2}2^p\sum_{k=1}^{\infty} b_k\int_{0}^{u}\sin^p kt\,dt, \]

\[ \left|\int_{0}^{1/n}\left[\int_{0}^{u}{}^{s}\Delta_t^p f(0)\,dt\right]du\right| \geqslant C_1(p)\sum_{k=1}^{n} b_k \int_{0}^{1/n}\left[\int_{0}^{u}\sin^p kt\,dt\right]du, \]

\[ \frac{1}{n^p}\sum_{k=1}^{n} k^p b_k \leqslant C(p)\left|\,n^2\int_{0}^{1/n}\left[\int_{0}^{u}{}^{s}\Delta_t^p f(0)\,dt\right]du\,\right|. \]

The theorem is proved.

Corollary. If \(b_k\downarrow 0\), then

\[ b_n = O\left( \left| n\int_{0}^{1/n}\left[\int_{0}^{u}{}^{s}\Delta_t^p f(0)\,dt\right]du \right| \right). \]

In particular, if \(f(x)\) has \(m\) derivatives, and \(f^{(m)}(x)\in \operatorname{Lip}\alpha\) \((0<\alpha<1)\) or \(Z\) \((\alpha=1)\), then

\[ b_k=O\left(\frac{1}{k^{1+m+\alpha}}\right). \]

For

\[ f(x)=\sum_{k=0}^{\infty} b_k\sin kx \]

with \(m=0,\;0=m<\alpha<1\), this fact was established by Lorentz \((^3)\).

Theorem 5. Let

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

where \(a_k\downarrow 0,\; b_k\downarrow 0\). In order that \(f(x)\) have a derivative of order \(p\) belonging to \(\operatorname{Lip}\alpha\) \((0<\alpha<1)\) or \(Z\) \((\alpha=1)\), it is necessary and sufficient that

\[ a_k=O\left(\frac{1}{k^{1+p+\alpha}}\right), \qquad b_k=O\left(\frac{1}{k^{1+p+\alpha}}\right). \]

The necessity of the condition is clear from the corollaries to Theorems 3 and 4. The sufficiency is proved analogously to the sufficiency in Theorem 2.

The author expresses his deep gratitude to Prof. I. P. Natanson for a number of valuable comments.

Leningrad State University
named after A. A. Zhdanov

Received
12 VI 1964

CITED LITERATURE

\(^1\) N. K. Bari, Trigonometric Series, Moscow, 1961.
\(^2\) I. P. Natanson, Constructive Function Theory, Moscow–Leningrad, 1949.
\(^3\) G. G. Lorentz, Math. Zs., 51, 135 (1948).