Abstract
Full Text
UDC 517.522
MATHEMATICS
P. P. KOROVKIN
AN EXTREMAL PROBLEM AND LINEAR METHODS OF SUMMATION OF FOURIER SERIES
(Presented by Academician V. I. Smirnov on 1 II 1967)
Let \(\{h_k\}_0^\infty\) be a sequence of real numbers. Denote by \(m_s(h_0,h_1,\ldots,h_s)\) the minimum of the function
\[ z_s=\sum_{k=0}^{s}(x_k^2+y_k^2) \]
subject to the constraint equations
\[ \sum_{k=0}^{s}(x_k^2-y_k^2)=h_0,\qquad 2\sum_{k=0}^{s-i}(x_kx_{k+i}-y_ky_{k+i})=h_i,\quad i=1,2,\ldots,s. \]
Theorem. If the series
\[ \sum_{i=0}^{\infty} h_i\cos ix=f(x) \]
converges uniformly on the real axis, then
\[ m(h_0,h_1,\ldots)=\lim_{s\to\infty}m_s(h_0,h_1,\ldots,h_s) =\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx. \]
Proof. Since
\[ u_s(x)=\left|\sum_{k=0}^{s}x_ke^{ikx}\right|^2 =\sum_{k=0}^{s}x_k^2+2\sum_{i=1}^{s}\sum_{k=0}^{s-i}x_kx_{k+i}\cos ix, \]
\[ v_s(x)=\left|\sum_{k=0}^{s}y_ke^{ikx}\right|^2 =\sum_{k=0}^{s}y_k^2+2\sum_{i=1}^{s}\sum_{k=0}^{s-i}y_ky_{k+i}\cos ix, \]
the constraint equations may be rewritten as
\[ u_s(x)-v_s(x)=\sum_{i=0}^{s}h_i\cos ix. \]
Taking into account the positivity of the trigonometric polynomials \(u_s(x)\) and \(v_s(x)\), we obtain
\[ \frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\sum_{i=0}^{s}h_i\cos ix\right|\,dx \le \frac{1}{2\pi}\int_{-\pi}^{\pi}[u_s(x)+v_s(x)]\,dx =\sum_{k=0}^{s}(x_k^2+y_k^2)=Z_s. \]
It follows from this inequality that
\[ m_s(h_0,h_1,\ldots,h_s)\ge \frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\sum_{i=0}^{s}h_i\cos ix\right|\,dx, \]
\[ \lim_{s\to\infty}m(h_0,h_1,\ldots,h_s)\ge \lim_{s\to\infty}\int_{-\pi}^{\pi}\left|\sum_{i=0}^{s}h_i\cos ix\right|\,dx =\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx. \tag{1} \]
To obtain the reverse relation, set
\[ f_+(x)=(f(x)+|f(x)|)/2. \]
The function \(f_+(x)\) is continuous and positive. Consequently, there exists a trigonometric polynomial \(\widetilde u_s(x)\) of order \(s\) such that the relations
\[ \text{1) }\ \widetilde u_s(x)>f_+(x); \qquad \text{2) }\ \widetilde u_s(x)-f_+(x)<\varepsilon,\ \ \varepsilon>0. \]
hold. Put
\[ \widetilde v_n(x)=\widetilde u_s(x)-\sum_{k=0}^{n} h_k\cos kt. \]
The sequence of functions \(\widetilde v_n(x)\) converges uniformly to the positive function \(\widetilde u_s(x)-f(x)\). Thus, for all sufficiently large indices \(n,\ n\ge s\), the inequalities
\[ 0\le \widetilde v_n(x)<\widetilde u_s(x)-f(x)+\varepsilon. \]
will hold.
By Fejér’s theorem \((^1)\), the positive trigonometric polynomials \(\widetilde u_s(x)\) and \(\widetilde v_n(x)\) can be written in the form
\[ \widetilde u_s(x)=\left|\sum_{k=0}^{s}\widetilde x_k e^{ikx}\right|^2 = \left|\sum_{k=0}^{n}\widetilde x_k e^{ikx}\right|^2, \qquad \widetilde x_k=0,\quad k>s, \]
\[ \widetilde v_n(x)=\left|\sum_{k=0}^{n}\widetilde y_k e^{ikx}\right|^2. \]
For these polynomials the connection equations are valid:
\[ \widetilde u_s(x)-\widetilde v_n(x)=\sum_{k=0}^{n} h_k\cos kx, \]
i.e.,
\[ \sum_{k=0}^{n}(\widetilde x_k^{\,2}-\widetilde y_k^{\,2})=h_0,\qquad 2\sum_{k=0}^{n-i}(\widetilde x_k\widetilde x_{k+i}-\widetilde y_k\widetilde y_{k+i})=h_i,\quad i=1,2,\ldots,n. \]
Putting \(f_-(x)=f(x)-f_+(x)\), we obtain
\[ \sum_{k=0}^{n}(\widetilde x_k^{\,2}+\widetilde y_k^{\,2}) = \frac{1}{2\pi}\int_{-\pi}^{\pi}[\widetilde u_s(x)+\widetilde v_n(x)]\,dx \le \]
\[ \le \frac{1}{2\pi}\int_{-\pi}^{\pi}[f_+(x)+\varepsilon+f_-(x)+2\varepsilon]\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx+3\varepsilon. \]
It follows that, for all sufficiently large indices \(n\), the inequality
\[ m_n(h_0,h_1,\ldots,h_n)\le \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx+3\varepsilon \]
will be true. Since \(\varepsilon>0\) is arbitrary, we have
\[ \varlimsup_{n\to\infty} m(h_0,h_1,\ldots,h_n)\le \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx, \tag{2} \]
and the theorem follows from relations (1) and (2).
Since every trigonometric polynomial can also be regarded as a uniformly convergent trigonometric series, we have
\[ m(h_0,h_1,\ldots,h_n,0,0,\ldots) = \frac{1}{2\pi}\int_{-\pi}^{\pi} \left|\sum_{k=0}^{n}h_k\cos kx\right|\,dx. \tag{3} \]
Relying on S. N. Bernstein’s theorems on the order of growth of the derivatives of a trigonometric polynomial and Jackson’s theorem on the order of approximation of functions by trigonometric polynomials, it is easy to show that, for
\[ |h_k| \leq M < \infty \]
\[ \left|m_{n^2}(h_0,h_1,\ldots,h_n,0,0,\ldots,0)-m(h_0,h_1,\ldots,h_n,0,0,\ldots)\right|\leq c . \]
The quantity \(c\) does not depend on the number \(n\).
By this equality the problem of estimating the integral (3), which is of importance in the theory of summation of Fourier series, is reduced to the algebraic problem of estimating the quantity \(m_{n^2}(h_0,\ldots,h_n,0,\ldots,0)\).
Corollary. If \(f(x)\) is an even function of class \(\mathcal L_p\), \(p>1\), and
\[ \sum_{k=0}^{\infty} h_k \cos kx \]
is its Fourier series, then
\[ \lim_{n\to\infty} m(h_0,h_1,\ldots,h_n,0,0,\ldots) = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx . \]
Moscow Automobile and Road Institute
Received
15 I 1967
CITED LITERATURE
- G. Pólya, G. Szegő, Problems and Theorems in Analysis, Part II, Moscow, 1956.