UDC 517.512.2

MATHEMATICS

Ya. L. Geronimus

SEVERAL REMARKS ON THE METHOD OF MULTIPLIERS

(Presented by Academician S. N. Bernstein, 9 VI 1965)

I. Let \(\Lambda=\{\lambda_k^{(n)}\}\) \((k=0,1,\ldots,n;\; n=1,2,\ldots)\) be a triangular matrix of real numbers, with

\[ \Delta_2\lambda_k^{(n)} = \lambda_k^{(n)}-2\lambda_{k+1}^{(n)}+\lambda_{k+2}^{(n)} \quad (k=0,1,\ldots,n-1); \qquad \lambda_0^{(n)}=1,\ \lambda_{n+1}^{(n)}=0. \tag{1} \]

In the theory of linear methods of summation of Fourier series, an important role is played by estimates for the mean value of the kernel, i.e. for the integral

\[ \frac{1}{\pi}\int_0^\pi |K_n(t)|\,dt, \qquad K_n(t)=\frac{\lambda_0^{(n)}}{2}+\sum_{k=1}^n \lambda_k^{(n)}\cos kt. \tag{2} \]

A simple upper estimate is given by

Theorem 1. If for each value of \(n\) the sequence of numbers \(\{\lambda_k^{(n)}\}_0^n\) is convex or concave, i.e.

\[ \varepsilon_n\Delta_2\lambda_k^{(n)}\geqslant 0 \quad (k=0,1,\ldots,n-1;\ n=1,2,\ldots), \qquad \varepsilon_n=\pm 1, \tag{3} \]

then the estimate holds

\[ \frac{1}{\pi}\int_0^\pi |K_n(t)|\,dt \leqslant \frac{1}{2}+C(n+1)|\lambda_n^{(n)}|; \tag{4} \]

in the particular case where \(\varepsilon_n=1,\ \lambda_n^{(n)}\geqslant 0\), one may take \(C=0\).

For the proof we apply Abel’s transformation twice to the series \(K_n(t)\); we obtain

\[ K_n(t)=\sum_{k=0}^{n-1}\Delta_2\lambda_k^{(n)}\cdot S_k(t) +\lambda_n^{(n)}S_n(t), \qquad S_k(t)=\frac{1}{2} \left( \frac{\sin \frac{k+1}{2}t}{\sin \frac{1}{2}t} \right)^2, \]

\[ \frac{1}{\pi}\int_0^\pi |K_n(t)|\,dt \leqslant \frac{1}{2}\sum_{k=0}^{n-1}(k+1)|\Delta_2\lambda_k^{(n)}| +\frac{1}{2}(n+1)|\lambda_n^{(n)}|; \]

from the obvious equality

\[ \lambda_0^{(n)} = \sum_{k=0}^{n-1}(k+1)\Delta_2\lambda_k^{(n)} + (n+1)\lambda_n^{(n)} \]

we obtain

\[ \frac{1}{\pi}\int_0^\pi |K_n(t)|\,dt \leqslant \frac{\varepsilon_n\lambda_0^{(n)}}{2} + \frac{n+1}{2}\{|\lambda_n^{(n)}|-\varepsilon_n\lambda_n^{(n)}\}, \]

whence (4) follows.

* L. Fejér (⁵) showed that in this case the kernel is nonnegative.

II. From Theorem 1 it follows: under condition (3), the additional condition

\[ |\lambda_n^{(n)}|=O(1/n) \tag{5} \]

is sufficient for the validity of the inequality

\[ \frac1\pi \int_0^\pi |K_n(t)|\,dt \leqslant C_1, \tag{6} \]

where \(C_1\) does not depend on \(n\).

Although the simple condition (5) is not necessary, it is nevertheless applicable in almost all known particular cases.

1) The Cesàro method \((C,r)\), \(r\geqslant 0\); in this case we have

\[ \lambda_n^{(n)}=n!\Gamma(r+1)/\Gamma(n+r+1)\simeq e^r\Gamma(r+1)/n^r, \]

i.e., condition (5) is applicable for \(r\geqslant 1\).

2) The Vallée-Poussin method

\[ \lambda_k^{(n)}= \begin{cases} 1 & (k=0,1,\ldots,n-p),\\ (n-k+1)/(p+1) & (k=n-p+1,\ldots,n); \end{cases} \]

we have \(\lambda_n^{(n)}=1/(p+1)\), and (5) is equivalent to the condition \(\liminf\limits_{n\to\infty} p/n>0\).

3) Let \(\lambda_k^{(n)}=\varphi(k/(n+1))\) \((k=0,1,\ldots,n)\), where \(\varphi(0)=1\), \(\varphi(1)=0\); the function \(\varphi(x)\) has on the interval \([0,1]\) a piecewise continuous second derivative that does not change sign on this interval; then

\[ \lambda_{n+1}^{(n)}-\lambda_n^{(n)} =-\lambda_n^{(n)} =\varphi(1)-\varphi\!\left(\frac{n}{n+1}\right) =\frac1{n+1}\varphi'(z), \]

\[ \frac{n}{n+1}<z<1,\qquad |\lambda_n^{(n)}|\leqslant \frac{|\varphi'(z)|}{n+1}, \]

i.e., condition (5) is fulfilled owing to the boundedness of \(|\varphi'(x)|\) in a neighborhood of the point \(x=1\). In particular, it is fulfilled in the method of S. N. Bernstein, where \(\varphi(x)=\cos \pi x/2\), and also in the case \(\varphi(x)=(1-x)^\alpha\), if \(\alpha\geqslant 1\), etc.

III. Let \(f(\theta)\in \mathcal L(-\pi,\pi)\),

\[ f(\theta)\sim \sum_{k=0}^{\infty}(a_k\cos k\theta+b_k\sin k\theta), \tag{7} \]

\[ U_n(f,\Lambda;\theta)=\frac{\lambda_0^{(n)}a_0}{2} +\sum_{k=1}^{n}\lambda_k^{(n)}(a_k\cos k\theta+b_k\sin k\theta). \]

Theorem 2. For the convergence

\[ \lim_{n\to\infty} U_n(f,\Lambda;\theta)=f(\theta) \tag{8} \]

at every Lebesgue point of the function \(f(\theta)\), the conditions (3), (5), and the additional condition

\[ \lim_{n\to\infty}\lambda_k^{(n)}=1\qquad (k=0,1,\ldots) \tag{9} \]

are sufficient.

The proof follows from the results of S. M. Nikol’skii \((^3)\).

IV. S. B. Stechkin \((^4)\) posed the question of finding the best constant in the inequality

\[ \left|\sum_{k=0}^{n}\frac{\lambda_k^{(n)}}{n-k+1}\right| \leqslant C\int_0^\pi |K_n(t)|\,dt . \tag{10} \]

Using our results (²) and the results of N. I. Akhiezer and M. G. Krein (¹), one can give an algorithm for solving this problem.

Theorem 3. The best value of the constant \(C\) in (10) is equal to the largest positive root of the equation

\[ \left| \begin{array}{cccc} \gamma_0 & \gamma_1 & \ldots & \gamma_n\\ \gamma_{-1} & \gamma_0 & \ldots & \gamma_{n-1}\\ \cdot & \cdot & \cdot & \cdot\\ \gamma_{-n} & \gamma_{-n+1} & \ldots & \gamma_0 \end{array} \right| =0,\qquad \gamma_{-k}=\overline{\gamma_k},\quad \gamma_0=\gamma+\overline{\gamma}, \tag{11} \]

where \(\gamma, \{\gamma_k\}_1^n\) are the first coefficients of the expansion

\[ \gamma+\sum_{k=1}^{n}\gamma_k z^k+O(z^{n+1}) = \exp\left\{\frac{i}{C}\sum_{k=0}^{n}\frac{z^k}{\,n-k+1\,}+O(z^{n+1})\right\}. \tag{12} \]

Kharkov
Aviation Institute

Received
4 VI 1965

REFERENCES CITED

¹ N. I. Akhiezer, M. G. Krein, Communications of the Kharkov Mathematical Society, ser. 4, 9, 9 (1934). ² Ya. L. Geronimus, C. R., 199, 1010 (1934). ³ S. M. Nikol’skii, Izv. Acad. Sci. USSR, Ser. Math., 12, 259 (1948). ⁴ S. B. Stechkin, UMN, 10, No. 1 (63), 159 (1955). ⁵ L. Fejér, Acta Lit. Acad. Sci. Univ. Franc. Joseph., 2, 75 (1924).