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V. V. ZHUK
ON SOME MODIFICATIONS OF THE CONCEPT OF THE MODULUS OF SMOOTHNESS AND THEIR APPLICATIONS
(Presented by Academician L. V. Kantorovich on 23 XI 1964)
1°. We shall use the following notation. A function \(f(x)\in L_{2\pi}\); \(\omega_k(\delta,f)_{L^p}\) is its modulus of smoothness of order \(k\) in the metric \(L^p\) \((L^\infty=C_{2\pi})\); \(E_n(f)_{L^p}\) denotes best approximations by trigonometric polynomials of degree not exceeding \(n\); \(\sigma(f)\) is the Fourier series; \(c>0\) is a finite constant depending only on those arguments which will be written out. Put
\[ {}^{s}\Delta_t^r f(x)=\sum_{k=0}^{r}(-1)^k C_r^k f[x+(r-2k)t], \]
\[ \sigma(f)=\frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx), \]
\[ F(x)=\sum_{k=1}^{\infty}\left(-\frac{b_k}{k}\cos kx+\frac{a_k}{k}\sin kx\right). \]
In a number of questions it is possible to characterize the structural properties of a function by the moduli of smoothness of the primitive (and not of the function itself!), and also by the quantities introduced below (the \(L\)-modulus of smoothness). This makes it possible to refine some well-known results.
2°. Absolute convergence of Fourier series. Numerous works are devoted to the question of absolute convergence. The fundamental theorems are those of S. N. Bernstein \((^1)\), pp. 217–223, 166–169. A detailed exposition of the results pertaining to this subject is given in the book of N. K. Bari \((^2)\).
Theorem 1. Let the sequence \(n_k\uparrow\infty\) be such that
\[ \sum_{z=k}^{\infty}\frac{(a_{n_z}^{\,2}+b_{n_z}^{\,2})}{z^2} \leq C_1\sum_{n=n_k}^{\infty}\frac{(a_n^{\,2}+b_n^{\,2})}{n^2}. \tag{1} \]
Then
\[ \sum_{k=1}^{\infty}(|a_{n_k}|+|b_{n_k}|) \leq \frac{2\sqrt{2C_1}}{\sqrt{\pi}} \sum_{k=1}^{\infty}\sqrt{k}\,E_{n_k-1}(F)_{L^2}. \]
Corollary 1. Let \(f(x)\in L_{2\pi}\). Then
\[ \sum_{k=1}^{\infty}(|a_k|+|b_k|) \leq \frac{2\sqrt{2}}{\sqrt{\pi}} \sum_{k=1}^{\infty}\sqrt{k}\,E_{k-1}(F)_{L^2}. \]
Corollary 2. If \(E_n(f)_L=O(1/n)\), then
\[ \sum_{k=1}^{\infty}(|a_k|+|b_k|) \leq C_2(r)\sum_{k=1}^{\infty} k^{-1/2} \left[\omega_k\left(\frac{1}{k},F\right)_C\right]^{1/2}. \]
There exist absolutely continuous functions satisfying the condition of Corollary 2 and not satisfying Zygmund’s condition.
([2], p. 614)
\[ \sum_{k=1}^{\infty} k^{-1}\sqrt{\omega_{1}(1/k,f)_{C}}<+\infty \]
and the stronger condition
\[ \sum_{k=1}^{\infty} k^{-1}\sqrt{\omega_{2}(1/k,f)_{C}}<+\infty . \]
Theorem 2. Let the sequence \(n_k\uparrow\infty\) satisfy condition (1), and let \(\Phi(u)\) be such that \(\Phi(0)=0\), \(\Phi(u)\) is increasing, \(\Phi(u)\) is convex upward, and
\[
\Phi(u_1\times u_2)\leq \Phi(u_1)\Phi(u_2).
\]
Then
\[ \sum_{k=1}^{\infty}\Phi\left(a_{n_k}^{\,2}+b_{n_k}^{\,2}\right) \leq \Phi(4)\,\Phi\left(\frac{C_1}{\pi}E_{n_{k-1}}^{2}(F)_{L^2}\right) + \]
\[ +2\sum_{k=1}^{\infty}\Phi(8k)\Phi\left(\frac{C_1}{\pi}\times E_{n_{k-1}}^{2}(F)_{L^2}\right). \]
Definition 1. The quantity
\[ \widetilde{L}_{p}^{\,r}(h,f)= \frac{1}{h}\sup_{0\leq u\leq h} \left\| \int_{0}^{u}{}^{s}\Delta_{t}^{r}f(x)\,dt \right\|_{L^{p}} \]
will be called the \(\widetilde{L}\)-modulus of smoothness of order \(r\) of the function \(f(x)\) in the metric \(L^{p}\).
Let us note a number of properties of the \(\widetilde{L}\)-modulus of smoothness:
-
If, as \(h\to0\), \(\widetilde{L}_{1}^{(r)}(h,f)=o(h^{2})\), then \(f(x)\sim\mathrm{const}\).
-
In order that \(f(x)\in L^{p}\), where \(1\leq p\leq\infty\), have an \((r-1)\)-st derivative belonging to \(\operatorname{Lip}(1,p)\), it is necessary and sufficient that
\[ L_{p}^{(r)}(h,f)=O(h^{2}). \] -
Let \(f(x)\in L^{2}\). In order that
\[ \omega_r(1/n,f)_{L^2}\sim \widetilde{L}_{2}^{(r)}(1/n,f), \tag{2} \]
it is necessary and sufficient that
\[
E_n|f|_{L^2}\leq C_3(r)\widetilde{L}_{2}^{(r)}(1/n,f).
\]
Remark. Equality (2) is not always fulfilled.
Theorem 3. Let \(f(x)\in L\) (\(r\) odd) and \(f(x)\in L^{2}\) (\(r\) even). If \(0<m<2\), then
\[ \sum_{k=n}^{\infty}\left(|a_k|^{m}+|b_k|^{m}\right) \leq C_4(r,m) \left\{ \sum_{k=n}^{\infty}k^{-m/2} \left[\widetilde{L}_{2}^{(r)}\left(\frac{1}{k},f\right)\right]^{m} + \left[\widetilde{L}_{2}^{(r)}\left(\frac{1}{n},f\right)\right]n^{1-m/2} \right\}. \]
Theorem 4. Let \(f(x)\in L\).
- If \(a_k\geq0\) \((k=1,2,\ldots)\), then* for \(r\) odd
\[ n^{-r}\sum_{k=1}^{n}a_k k^{r-1} \leq C_5(r)\widetilde{L}_{C}^{(r)}\left(\frac{1}{n},F\right). \]
- If \(b_k\geq0\) \((k=1,2,\ldots)\), then for \(r\) even
\[ n^{-r}\sum_{k=1}^{n}b_k k^{r-1} \leq C_6(r)\widetilde{L}_{C}^{(r)}\left(\frac{1}{n},F\right). \]
- If \(a_k\geq0\) \((k=1,2,\ldots)\), then
\[ \sum_{k=n}^{\infty}a_k \leq C_7\sum_{k=n}^{\infty}\omega_r\left(\frac{1}{k},F\right)_{C}. \]
- If \(b_k\geq0\) \((k=1,2,\ldots)\), then
\[ \sum_{k=n}^{\infty}b_k \leq C_8\sum_{k=n}^{\infty}\omega_r\left(\frac{1}{k},F\right)_{C}. \]
\[ \text{* No restrictions are imposed on } b_k. \]
3°. Multipliers of uniform convergence. (A survey of known results is given in the work of F. I. Kharshiladze \((^3)\).) Let
\[ \Lambda_n(t)=\lambda_0/2+\sum_{k=1}^{n}\lambda_k\cos kt . \]
Theorem 5. If the conditions
\[ \int_{0}^{2\pi}\left|\sum_{\nu=0}^{n}\Lambda_\nu(t)\right|dt=O(n),\qquad E_{[(n+1)/2]}(F)_C\int_{0}^{2\pi}|\Lambda_n(t)|dt=O\left(\frac1n\right), \]
are satisfied, then for a continuous \(f(x)\) the numbers \(\lambda_k\) are multipliers of uniform convergence.
Remark. Already for \(\lambda_k\equiv 1\) \((k=0,1,2,\ldots)\) there exist functions satisfying the conditions of Theorem 5, whereas the conditions of the theorems of Bojanic \((^4)\) and Kharshiladze \((^3)\) are not fulfilled. An example may be
\[ f(x)=\sum_{n=1}^{\infty}\frac{\cos 2^n x}{n^{1+\alpha}}\quad (0<\alpha<1)\quad ((^2),\ \text{pp. }294\text{--}296). \]
4°. The concept of the \(\widetilde L\)-modulus of smoothness has applications in the theory of singular integrals. Relying on the properties of the \(\widetilde L\)-modulus, one can prove the following theorems.
Theorem 6. The following relations are equivalent:
\[ f(x)\in \operatorname{Lip}(1,p), \]
\[ \left\|\frac{3}{2\pi n(2n^2+1)} \int_{-\pi}^{\pi}|f(x+t)-f(x)| \left(\frac{\sin nt/2}{\sin t/2}\right)^4dt\right\|_{L^p} =O\left(\frac1n\right), \]
\[ \left\|\frac{(2n)!!}{(2n+1)!!}\frac1{2\pi} \int_{-\pi}^{\pi}|f(x+t)-f(x)|\cos^{2n}\frac t2\,dt\right\|_{L^p} =O\left(\frac1{\sqrt n}\right). \]
Theorem 7. In order that the relation
\[ \left\|\frac{(2n)!!}{(2n-1)!!} \int_{0}^{\pi}|f(x+t)-2f(x)+f(x-t)|\cos^{2n}\frac t2\,dt\right\|_{L^p} =O\left(\frac1{\sqrt{n^\alpha}}\right), \]
hold, where \(0<\alpha\le 2\), it is necessary and sufficient that, as \(t\to0\),
\[ \omega_2(t,f)=O(t^\alpha). \]
Theorem 8. In order that, as \(r\to1-0\),
\[ \left\|\frac1{2\pi}\int_{0}^{\pi}|f(x+t)-2f(x)+f(x-t)| \frac{1-r^2}{1-2r\cos t+r^2}\,dt\right\|_{L^p} =O((1-r)^\alpha), \]
where \(0<\alpha<1\), it is necessary and sufficient that \(f(x)\in \operatorname{Lip}(\alpha,p)\).
5°. Generalized methods of summation of Fourier series in a Hilbert space. Let \(H=\{x\}\) be a Hilbert space, \(\{x_n\}\) \((n=1,2,\ldots)\) a complete orthonormal system in \(H\),
\[ E_n(x)_H=\left[\sum_{k=n+1}^{\infty}(x,x_k)^2\right]^{1/2}; \]
to each element \(x\in H\) there is assigned the series
\[ U(x,\xi)=\sum_{k=1}^{\infty}\gamma_k(\xi)(x,x_k)x_k, \]
where the functions \(\gamma_k(\xi)\) are given on some set with a limit point \(w\). We assume that \(\gamma_1(\xi)=1\) and
\[ \sum_{k=1}^{\infty}\gamma_k^2(\xi)(x,x_k)^2<+\infty, \]
at least for values ...
functions \(\xi\) close to \(w\). In what follows, the summability method is considered only for such \(\xi\). M. F. Timan \(\left((^5),\right.\) p. 351; \(\left.(^6)\right)\) proved that
\[ \omega_r\left({1\over n},f\right)_{L^2}\sim {1\over n^r}\left[\sum_{k=1}^{n} k^{2r-1}E_k^2(f)_{L^2}\right]^{1/2}. \]
Definition 2. The quantity
\[ \widetilde{\omega}_r\left({1\over n},x\right)_H ={1\over n^r}\left[\sum_{k=1}^{n} k^{2r-1}E_k^2(x)_H\right]^{1/2} \]
will be called the \(\widetilde{\omega}\)-modulus of smoothness of order \(r\) of the element \(x\) in the space \(H\).
Definition 3. Let \(\varphi(\xi)\) be such that \(\lim_{\xi\to 0}\varphi(\xi)=0\), and for all \(x\in H\) \((x\in \{Cx_1\})\),
\[ \|x-U(x,\xi)\|_H \ge C_9(x)\varphi(\xi) \]
and there exists \(x\in H\) \((x\in \{Cx_1\})\) for which
\[ \|x-U(x,\xi)\|_H \le C_{10}(x)\varphi(\xi). \tag{3} \]
Then we shall say that the summability method is saturated in \(H\) with order \(\varphi(\xi)\), and call the class of saturation the set of \(x\in H\) for which (3) is fulfilled.
Theorem 9. Let \(x\in H,\ r>0\). Then the series
\[ \sum_{k=1}^{\infty} k^{2r-1}E_k^2(x)_H,\qquad \sum_{k=1}^{\infty} k^{2r}(x,x_k)^2,\qquad \sum_{k=1}^{\infty} k^{2r-1}\widetilde{\omega}_{r+1}^2\left({1\over k},x\right)_H \]
converge or diverge simultaneously.
Remark 1. Let \(f(x)\in L_{2\pi}^2\). Then the series \(\sum_{k=1}^{\infty}(a_k^2+b_k^2)k^{2r}\) \((r\ge 1)\) and the integral
\[ \int_0^{2\pi}\int_0^{2\pi}{\left|{}^s\Delta_t^{\,r+1}f(x)\right|^2\over t^{2r+1}}\,dt\,dx \]
converge or diverge simultaneously.
Remark 2. For \(r=0\), the theorem and Remark 1 do not hold \(\left((^2),\text{ pp. }347,399\right)\).
Theorem 10. If \(x\in H\), then
\[ \|x-U(x,\xi)\|_H =\left[\sum_{k=1}^{\infty}E_k^2(x)_H\left[(1-\gamma_{k+1}(\xi))^2-(1-\gamma_k(\xi))^2\right]\right]^{1/2}. \]
Corollary. Let \(\gamma_k(n)=1-((k-1)/n)^p\), if \(k=1,\ldots,n\), and \(\gamma_k(n)=0\) for \(k>n\), where \(p>0\) is any real number. Then
\[ {1\over n^p}\left[E_1^2(x)_H+2p\sum_{k=2}^{n}(k-1)^{2p-1}E_k^2(x)_H\right]^{1/2}\le \]
\[ \le \|x-U^{(p)}(x,n)\|_H \le {\sqrt{2p}\over n^p}\left[\sum_{k=1}^{n}k^{2p-1}E_k^2(x)_H\right]^{1/2}. \]
The summability method is saturated with order \(1/n^p\). The class of saturation is the set of elements for which
\[ \sum_{k=1}^{\infty} k^{2p-1}E_k^2(x)_H<+\infty. \]
Relying on Theorem 10, it is not difficult to obtain analogous results for other summability methods as well.
Leningrad State University
named after A. A. Zhdanov
Received
16 XI 1964
CITED LITERATURE
- S. N. Bernstein, Collected Works, vol. 1, Publishing House of the Academy of Sciences of the USSR, 1952.
- N. K. Bari, Trigonometric Series, Moscow, 1961.
- F. I. Kharshiladze, Proceedings of the Tbilisi Mathematical Institute, 26, 121 (1959).
- R. Bajanic, Publ. Inst. Math. Acad. Serbe Sci., 10, 153 (1956).
- A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.
- M. F. Timan, Abstract of doctoral dissertation, Tbilisi, 1962.