Mathematics

I. M. Ganzburg

A Generalization of Some Results of S. M. Nikol’skii and A. F. Timan

(Presented by Academician S. N. Bernstein on 27 IV 1957)

Let \(MH^{(\alpha)}\) be the class of functions \(f(x)\), defined on the interval \([-1,1]\) and satisfying the condition

\[ |f(x')-f(x'')|\leq M|x'-x''|^\alpha \qquad (0<\alpha\leq 1), \tag{1} \]

for arbitrary \(x',x''\in[-1,1]\), and let \(s_n(f,x)\) be the partial sum of the first \(n+1\) terms of the P. L. Chebyshev series for \(f(x)\), i.e.

\[ s_n(f,x)=\frac{a_0}{2}+\sum_{p=1}^{n} a_p\cos p\arccos x; \]

\[ a_p=\frac{2}{\pi}\int_{-1}^{1}\frac{f(u)\cos p\arccos u}{\sqrt{1-u^2}}\,du. \tag{2} \]

In the theory of approximation of functions on a finite interval, an important role is played by the sums

\[ \sigma_{n,k}(f,x)=\frac{1}{k+1}\sum_{\nu=n-k}^{n}s_\nu(f,x) \qquad (n=0,1,2,\ldots;\ 0\leq k\leq n), \]

which are truncated arithmetic means of sums of P. L. Chebyshev series.

Of interest is the problem of determining the asymptotic behavior of the upper bound of the deviations of functions \(f(x)\) from their sums \(\sigma_{n,k}(f,x)\), distributed over the class \(H^{(\alpha)}\), i.e. the asymptotic estimate of the quantity

\[ \mathcal{E}_n(MH^{(\alpha)},x)=\sup_{f\in MH^{(\alpha)}}|f(x)-\sigma_{n,k}(f,x)|. \]

For \(k=0\) and \(\alpha=1\) this problem was solved by S. M. Nikol’skii \((^1)\), and for \(k=0\) and \(0<\alpha<1\) by A. F. Timan \((^3)\)*.

In the present note an asymptotic expression is established for \(\mathcal{E}_n(MH^{(\alpha)},x)\) \((0<\alpha\leq 1)\) uniformly on the segment \([-1,1]\) for arbitrary \(0\leq k\leq n\) \((0<\alpha<1)\) and \(0\leq k\leq n^\theta,\ 0\leq\theta<1\) \((\alpha=1)\).

Theorem. As \(n\to\infty\), the following asymptotic equality holds:

\[ \mathcal{E}_n\equiv \mathcal{E}_n(MH^{(\alpha)},x) = \frac{2^{\alpha+1}M}{\pi^2 n^\alpha} \left(\sqrt{1-x^2}\right)^\alpha \int_{0}^{\pi/2} t^\alpha\sin t\,dt \cdot \ln\frac{n+1}{k+1} + O\!\left(\frac{1}{n^\alpha}\right), \tag{3} \]

where \(O(1)\) denotes a quantity uniformly bounded with respect to all \(x\in[-1,1]\) and all \(k\) \((0\leq k\leq n\) for \(0<\alpha<1\) and \(0\leq k\leq n^\theta\) \((\theta<1)\) for \(\alpha=1)\).

Proof. Put \(M=1\). We first consider the case

* The case \(\alpha=0\) was studied by L. M. Abramov \((^5)\) (for special cases see \((^2,^4)\)).

\(0<\alpha<1\). It is not difficult to see that the least upper bound \(\mathcal E_n(H^{(\alpha)},x)\) coincides with the least upper bound \(\mathcal E_n(\widetilde H^{(\alpha)},y)\), extended to the class \(\widetilde H^{(\alpha)}\) of even functions \(\varphi(u)\) of period \(2\pi\) satisfying the conditions

\[ |\varphi(u')-\varphi(u'')|\le |\cos u'-\cos u''|^\alpha, \qquad u',u''\in[0,\pi],\quad \varphi(y)=0\quad (y=\arccos x). \tag{4} \]

Hence

\[ \begin{aligned} \mathcal E_n &=\sup_{\varphi\in\widetilde H^{(\alpha)}}\left| \frac{1}{2\pi(k+1)} \int_0^\pi \left\{ \frac{\sin\frac{2n+1-k}{2}(u+y)\cdot\sin\frac{k+1}{2}(u+y)} {\sin^2\frac{u+y}{2}} \right.\right. \\ &\hspace{3.5cm}\left.\left. + \frac{\sin\frac{2n+1-k}{2}(u-y)\cdot\sin\frac{k+1}{2}(u-y)} {\sin^2\frac{u-y}{2}} \right\}du \right| =\sup_{\varphi\in\widetilde H^{(\alpha)}}|I(\varphi)|. \end{aligned} \tag{5} \]

Without loss of generality, we may assume that \(0\le x\le 1\).

From the relations

\[ \int_0^\pi \varphi(t) \left[ \frac{1}{\sin^2\frac{t\pm y}{2}} -\frac{4}{(t\pm y)^2} \right] (t\pm y)^2 R_{m,k}(t\pm y)\,dt = O\left(\frac1{n^\alpha}\right), \]

\[ \int_y^{\pi+y}\varphi(t-y)R_{m,k}(t)\,dt + \int_y^{\pi-y}\varphi(t+y)R_{m,k}(t)\,dt = O\left(\frac1{n^\alpha}\right), \]

where

\[ R_{m,k}(t)= \frac{2\sin\frac{m+1}{2}t\cdot\sin\frac{k+1}{2}t} {\pi(k+1)t^2}, \qquad m=2n-k, \]

there follows the validity of the equality

\[ I(\varphi)= \int_0^y[\varphi(y-t)+\varphi(y+t)]R_{m,k}(t)\,dt + O\left(\frac1{n^\alpha}\right). \tag{6} \]

Introducing the notation

\[ y=\frac{2r+1}{m+1}\pi+\delta, \]

where \(r\) takes natural values and \(0\le\delta<2\pi/(m+1)\), after a number of transformations we obtain from (6)

\[ \begin{aligned} I(\varphi) &= \frac{(m+1)^2}{2\pi^3(k+1)} \sum_{i=1}^{r} (-1)^{i-1} \frac{\sin\left(\frac{k+1}{m+1}i\pi\right)}{i^2} \\ &\quad\times \int_0^{\pi/(m+1)} \sin\frac{m+1}{2}u \left[ \varphi\left(y+u-\frac{2i\pi}{m+1}\right) + \varphi\left(y-u+\frac{2i\pi}{m+1}\right) \right. \\ &\hspace{4.3cm}\left. - \varphi\left(y-u-\frac{2i\pi}{m+1}\right) - \varphi\left(y+u+\frac{2i\pi}{m+1}\right) \right]du + O\left(\frac1{n^\alpha}\right). \end{aligned} \]

Since \(\varphi(t)\) belongs to the class \(\widetilde H^{(\alpha)}\) (4), we have

\[ \begin{aligned} |I(\varphi)| &\le \frac{2^{\alpha-1}(m+1)^2}{\pi^3(k+1)} \sum_{i=1}^{r} \frac{\left|\sin\frac{k+1}{m+1}i\pi\right|}{i^2} \left[ \sin^\alpha\left(y-\frac{2i\pi}{m+1}\right) + \sin^\alpha\left(y+\frac{2i\pi}{m+1}\right) \right] \\ &\quad\times \int_0^{\pi/(m+1)} \sin\frac{m+1}{2}u\cdot(\sin u)^\alpha\,du + O\left(\frac1{n^\alpha}\right) \end{aligned} \tag{7} \]

uniformly for all \(\varphi(t)\in\widetilde H^{(\alpha)}\).

On the other hand, one can indicate a function \(\varphi_n(x)\in \widetilde H^{(\alpha)}\) for which the integral \(I(\varphi_n)\), up to a term of order \(O\!\left(\frac1{n^\alpha}\right)\), coincides with the right-hand side of (7).

Indeed, define on \([0,\pi]\) an even function of period \(2\pi\), \(\varphi(t)\equiv\varphi_n(t)\), taking the following values:

\[ \begin{array}{ll} \beta_r, & 0\le t\le \delta+\dfrac{\pi}{m+1}; \\[1.2em] (-1)^i 2^{\alpha-1}\sin^\alpha\!\left(y-\dfrac{2i+1}{m+1}\pi\right)\times \sin^\alpha\!\left(y-t-\dfrac{2i\pi}{m+1}\right)+\beta_i, & y-\dfrac{2i+1}{m+1}\pi\le t\le y-\dfrac{2i\pi}{m+1}, \\[-0.2em] & i=r-1,\ r-2,\ldots,1; \\[1.2em] (-1)^{i-1}2^{\alpha-1}\sin^\alpha\!\left(y-\dfrac{2i+1}{m+1}\pi\right)\times \sin^\alpha\!\left(\dfrac{2i\pi}{m+1}+t-y\right)+\beta_i, & y-\dfrac{2i\pi}{m+1}\le t\le y-\dfrac{2i-1}{m+1}\pi, \\[-0.2em] & i=r,\ r-1,\ldots,2; \\[1.2em] 0, & y-\dfrac{2\pi}{m+1}\le t\le y+\dfrac{4\pi}{m+1}; \\[1.2em] (-1)^{i-1}2^{\alpha-1}\sin^\alpha\!\left(y+\dfrac{2i-1}{m+1}\pi\right)\times \sin^\alpha\!\left(\dfrac{2i\pi}{m+1}-t+y\right)+\gamma_i, & y+\dfrac{2i-1}{m+1}\pi\le t\le y+\dfrac{2i\pi}{m+1}, \\[-0.2em] & i=3,4,\ldots,\mu; \\[1.2em] (-1)^i2^{\alpha-1}\sin^\alpha\!\left(y+\dfrac{2i-1}{m+1}\pi\right)\times \sin^\alpha\!\left(t-\dfrac{2i\pi}{m+1}-y\right)+\gamma_i, & y+\dfrac{2i\pi}{m+1}\le t\le y+\dfrac{2i+1}{m+1}\pi, \\[-0.2em] & i=2,3,\ldots,\mu-1; \\[1.2em] 0, & y+\dfrac{2\mu\pi}{m+1}\le t\le y+\dfrac{2(\mu+2)\pi}{m+1}; \\[1.2em] (-1)^{i-1}2^{\alpha-1}\sin^\alpha\!\left(y+\dfrac{2i+1}{m+1}\pi\right)\times \sin^\alpha\!\left(\dfrac{2i\pi}{m+1}-t+y\right)+\delta_i, & y+\dfrac{2i-1}{m+1}\pi\le t\le y+\dfrac{2i\pi}{m+1}, \\[-0.2em] & i=\mu+3,\ \mu+4,\ldots,r; \\[1.2em] (-1)^i2^{\alpha-1}\sin^\alpha\!\left(y+\dfrac{2i+1}{m+1}\pi\right)\times \sin^\alpha\!\left(t-y-\dfrac{2i\pi}{m+1}\right)+\delta_i, & y+\dfrac{2i\pi}{m+1}\le t\le y+\dfrac{2i+1}{m+1}\pi, \\[-0.2em] & i=\mu+2,\ \mu+3,\ldots,r-1; \\[1.2em] \delta_r, & y+\dfrac{2r\pi}{m+1}\le t\le \pi, \end{array} \]

where \(\mu\) is the greatest even number for which still

\[ y+\frac{2\mu\pi}{m+1}<\frac{\pi}{2}; \]

\(\beta_i,\gamma_i\) and \(\delta_i\) are constants ensuring the continuity of the function \(\varphi_n(t)\)
\[ (\beta_1=\gamma_2=\gamma_\mu=\delta_{\mu+2}=0). \]

The function \(\varphi_n(t)\) thus introduced belongs to \(\widetilde H^{(\alpha)}\) and indeed turns inequality (7) into an asymptotic equality.

Thus, if one also takes into account the relations

\[ (\sin u)^\alpha-u^\alpha=O(u^{3\alpha}),\qquad \sin^\alpha\!\left(y\pm \frac{2i\pi}{m+1}\right)-(\sin y)^\alpha =O\!\left(\frac{i^\alpha}{m^\alpha}\right), \]

we find

\[ \mathcal{E}_n= \frac{(2\sin y)^\alpha}{(m+1)^\alpha}\, \frac{2^{\alpha+1}(m+1)}{\pi^3(k+1)} \sum_{i=1}^{r} \frac{\left|\sin \frac{k+1}{m+1} i\pi\right|}{i^2} \int_0^{\pi/2} t^\alpha \sin t\,dt +O\!\left(\frac{1}{n^\alpha}\right). \tag{8} \]

From (8), after some transformations, it follows that

\[ \mathcal{E}_n= \frac{2^{\alpha+1}\left(\sqrt{1-x^2}\right)^\alpha}{\pi^2 n^\alpha} \int_0^{\pi/2} t^\alpha \sin t\,dt\cdot \ln \frac{n+1}{k+1} +O\!\left(\frac{1}{n^\alpha}\right). \tag{9} \]

In the case \(\alpha=1\), the class \(H^{(1)}\) coincides with the class \(W\) of functions \(f(x)\) having almost everywhere on \([-1,1]\) a bounded derivative \(f'(x)\), \(|f'(x)|\leq 1\). In this case the question reduces to an asymptotic estimate of the upper bound of the integral

\[ I^*=\frac{1}{\pi(k+1)} \left| \int_0^\pi \left\{ \sin t \sum_{\nu=n-k}^{n} \left[D_{\nu}^{(1)}(t-y)+D_{\nu}^{(1)}(t+y)\right] \right\} f'(\cos t)\,dt \right|, \]

where

\[ D_{\nu}^{(1)}(u)=\sum_{i=\nu+1}^{\infty}\frac{\sin iu}{i}, \]

and the upper bound is extended to the class \(W\).

Performing Abel’s transformation twice in the kernels \(D_{\nu}^{(1)}(t\pm y)\) and taking into account some known estimates, we obtain the relation

\[ I^*= \frac{1}{\pi(k+1)} \left| \sum_{\nu=n-k}^{n} \frac{1}{\nu+1} \left\{ \int_0^{y-1/n} + \int_{y+1/n}^{\pi} \left[ \frac{\cos \frac{2\nu+1}{2}(t-y)}{\sin \frac{t-y}{2}} + \frac{\cos \frac{2\nu+1}{2}(t+y)}{\sin \frac{t+y}{2}} \right] \sin t\cdot f'(\cos t)\,dt \right\} \right| +O\!\left(\frac{1}{n}\right), \]

where, in the case \(y<1/n\), the limits of integration \(y-1/n\) and \(y+1/n\) are both replaced by \(1/n\).

After a series of estimates we arrive at the asymptotic equality

\[ \mathcal{E}_n(H^{(1)},x)= \frac{4}{\pi^2 n}\sqrt{1-x^2}\, \ln \frac{n+1}{k+1} +O\!\left(\frac{1}{n}\right). \tag{10} \]

Combining (9) and (10), we obtain relation (3) for arbitrary \(\alpha\) \((0<\alpha\leq 1)\).

I express my gratitude to Prof. A. F. Timan for proposing the problem and for discussing the results.

Dnepropetrovsk State University
named after the 300th anniversary of the reunification of Ukraine with Russia

Received
3 I 1957

CITED LITERATURE

\(^{1}\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 10, 295 (1946).
\(^{2}\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 4, 509 (1940).
\(^{3}\) A. F. Timan, DAN, 77, No. 6, 969 (1951).
\(^{4}\) A. F. Timan, DAN, 61, No. 6, 989 (1948).
\(^{5}\) L. M. Abramov, DAN, 98, 173 (1954).