MATHEMATICS

S. I. RABINOVICH

ON THE QUESTION OF EXTREMAL FUNCTIONS IN CERTAIN PROBLEMS OF APPROXIMATION THEORY

(Presented by Academician S. L. Sobolev on 25 IV 1958)

Let (W^{(r)}H^{(\alpha)}) be the class of (2\pi)-periodic functions having a derivative of order (r) satisfying a Lipschitz condition of degree (\alpha) with constant 1, and let (\lambda_k^{(n)}) ((k=0,1,\ldots,n+1;\ \lambda_0^{(n)}=1;\ \lambda_{n+1}^{(n)}=0)) be a triangular matrix of numbers satisfying the conditions

[
\mu_k^{(n)} \leqslant \mu_{k+1}^{(n)}, \qquad \Delta^2 \mu_k^{(n)} \geqslant 0,
\tag{1}
]

where

[
\mu_k^{(n)}=\frac{1-\lambda_k^{(n)}}{k^2}.
]

For each function (f(x)\in W^{(r)}H^{(\alpha)}) we form the sequence of trigonometric polynomials of order (n)

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

where (a_k) and (b_k) are the Fourier coefficients of the function (f(x)), and we investigate the behavior of the absolute value of the deviation (f(x)-U_n(f;\ x;\ \lambda)).

A. F. Timan showed ({}^{(3)}) that for any function (f(x)\in W^{(r)}H^{(\alpha)}), where (r\geqslant 0) is any integer and (0\leqslant\alpha<1), as (n\to\infty) the asymptotic inequality

[
|f(x)-U_n(f;\ x;\ \lambda)|\leqslant
]

[
\leqslant
\frac{2^{\alpha+1}}{\pi^2 n^\alpha}
\int_{0}^{\pi/2} t^\alpha \sin t\,dt\cdot
\left|
\sum_{k=1}^{n}\frac{\mu_k^{(n)}}{n-k+1}
-\frac{\ln n}{n^r}
\right|
+
O!\left(\frac{1}{n^{r+\alpha}}\right)
\tag{2}
]

holds (in the case (\alpha=0) the right-hand side must be doubled), and moreover for every (n) there exists a function (\varphi_n(x)) for which inequality (2) becomes an asymptotic equality.

However, the question remains open: does there exist in the class (W^{(r)}H^{(\alpha)}) a function (f(x)), independent of (n), for which inequality (2) would become an asymptotic equality at least along some sequence of values of (n). Such a question was first considered by S. M. Nikol’skii ({}^{(2)}) (as applied to polynomials of best approximation). The purpose of the present work is to solve this question for linear methods of approximation.

Theorem 1. If the matrix of numbers (\lambda_k^{(n)}) satisfies conditions (1), then for each value (x_0\in[-\pi;\pi]) there exists a function (f_0(x)\in W^{(r)}H^{(\alpha)}) ((0\leqslant\alpha<1,\ r\geqslant0) an integer), for which the inequalities hold

[
\left| f_0(x_0)-U_{n_i}(f_0;x_0;\lambda)\right| >
]

[

\frac{2^{\alpha+1}}{\pi^2 n_i^\alpha}\int_0^{\pi/2} t^\alpha \sin t\,dt\cdot
\left|\sum_{k=1}^{n_i}\frac{\mu_k^{(n_i)}}{n_i-k+1}-\frac{\ln n_i}{n_i^r}\right|(1-\varepsilon_{n_i}),
\tag{3}
]

where (n_i) ((i=1,2,\ldots)) is a certain increasing sequence of natural numbers and (\varepsilon_{n_i}\to 0) as (i\to\infty).

For the cases (\lambda_k^{(n)}=1) ((k=0,1,\ldots,n)), (r=0), (0<\alpha\leqslant 1), and (\lambda_k^{(n)}=1) ((k=0,1,\ldots,n)), (r=1,2,\ldots), (\alpha=0), this result was obtained by G. Ya. Doronina ((^1)).

The proof of Theorem 1 is based on the following lemma.

Lemma. If the matrix of numbers (\lambda_k^{(n)}) satisfies conditions (1), then for every (0\leqslant \alpha<1) and integer (r\geqslant 0) the relations

[
\frac{1}{\pi}\int_a^\pi f(t)L_{nr}(t)\,dt
=
O\left(\frac{1}{n^{r+\alpha}}\right),
]

[
\frac{1}{\pi}\int_0^{C/n} f(t)L_{nr}(t)\,dt
=
O\left(\frac{1}{n^{r+\alpha}}\right),
]

hold, where (0<a\leqslant \pi) and (C) are fixed numbers;

[
L_{nr}(t)=\sum_{k=0}^n \mu_k^{(n)}\cos\left(kt+\frac{r\pi}{2}\right)
+
\sum_{k=n+1}^{\infty}\frac{\cos(kt+r\pi/2)}{k^r}.
]

The equalities (4) hold uniformly over all functions (f(x)\in W^{(r)}H^{(\alpha)}).

For the proof of the theorem, consider the function (\varphi_n(x)), defined on ([0,\pi]) by the equalities

[
\varphi_n(x)=
\begin{cases}
0, & \text{if } 0\leqslant x\leqslant x_1^{(n)};\
& x_{n-1}^{(n)}\leqslant x\leqslant \pi \quad \text{for even } n,\
& x_{n-2}^{(n)}\leqslant x\leqslant \pi \quad \text{for odd } n;\[4pt]
(-1)^\nu 2^{\alpha-1}(x-x_\nu^{(n)})^\alpha, & \text{if } x_\nu^{(\alpha)}\leqslant x\leqslant x_{\nu+1/2}^{(n)},\
& \nu=1,2,\ldots,n-2 \quad \text{for even } n,\
& \nu=1,2,\ldots,n-3 \quad \text{for odd } n;\[4pt]
(-1)^\nu 2^{\alpha-1}(x_{\nu+1}^{(\alpha)}-x)^\alpha, & \text{if } x_{\nu+1/2}^{(n)}\leqslant x\leqslant x_{\nu+1}^{(n)},
\end{cases}
]

where

[
x_\nu^{(n)}=\frac{\nu+l-r/2}{n}\pi,
]

and (l) is the smallest integer greater than (r/2).

On the basis of the lemma just stated, one establishes the possibility of choosing an increasing sequence of natural numbers ({n_i}) such that, uniformly over all functions (\varphi(x)\in W^{(r)}H^{(\alpha)}), the inequalities

[
\frac{1}{\pi}\left|\int_0^{x_1^{(n_i-1)}} \varphi_{n_i}(t)L_{n_i r}(t)\,dt\right| >
]

[

\frac{2^\alpha}{\pi^2 n_i^\alpha}\int_0^{\pi/2} t^\alpha \sin t\,dt\cdot
\left|\sum_{k=1}^{n_i}\frac{\mu_k^{(n_i)}}{n_i-k+1}-\frac{\ln n_i}{n_i^r}\right|
\left(1-\frac{\varepsilon_{n_i}}{3}\right);
]

[
\frac{1}{\pi}\left|\int_{x_1^{(n_i-1)}}^{\pi}\varphi(t)L_{n_i r}(t)\,dt\right|
<
\frac{2^\alpha}{\pi^2 n_i^\alpha}\int_0^{\pi/2} t^\alpha\sin t\,dt\cdot
\left|\sum_{k=1}^{n_i}\frac{\mu_k^{(n_i)}}{n_i-k+1}-\frac{\ln n_i}{n_i^r}\right|
\frac{\varepsilon_{n_i}}{3},
]

[
\frac{1}{\pi}\left|\int_0^{x_1^{(n_i)}}\varphi(t)L_{n_i r}(t)\,dt\right|
<
\frac{2^\alpha}{\pi^2 n_i^\alpha}\int_0^{\pi/2} t^\alpha\sin t\,dt
\left|\sum_{k=1}^{n_i}\frac{\mu_k^{(n_i)}}{n_i-k+1}-\frac{\ln n_i}{n_i^r}\right|
\frac{\varepsilon_{n_i}}{3},
]

where ({\varepsilon_n}) is a certain null-sequence satisfying the condition

[
\overline{\lim}{n\to\infty}\varepsilon_n
\left|\sum\right|}^{n}\frac{\mu_k^{(n)}}{n-k+1}-\frac{\ln n}{n^r
=\infty
]

and the (2\pi)-periodic function (\varphi_r^*(x)) is considered, defined on ([-\pi,\pi]) by the equalities

[
\varphi_r^*(x)=
\begin{cases}
0, & \text{if } x_1^{(n_i)}\le x\le x_{\nu_0}^{(n_i)};\
\varphi_{n_i}(x), & \text{if } x_{\nu_0}^{(n_i)}\le x\le \beta_{i-1}\quad (i=1,2,\ldots);\
0, & \text{if } \beta_{i-1}\le x\le x_1^{(n_i-1)};
\end{cases}
]

[
\varphi_r^*(0)=0;
]

[
\varphi_r^(-x)=\varphi_r^(x),\quad \text{if } r \text{ is even};
]

[
\varphi_r^(-x)=-\varphi_r^(x),\quad \text{if } r \text{ is odd}.
]

Here (\beta_{i-1}) is the zero of the function (\varphi_{n_i}(x)) nearest on the left to (x_1^{(n_i-1)}) such that
(\varphi_{n_i}(\beta_{i-1}-\pi/2n_i)); (\nu) is the smallest odd number satisfying the condition
(\nu_0>2^{1/\alpha-1}+1).

The function (f_0(x)), defined for the given (x_0) by the relation

[
f_0^{(r)}(x_0+x)=\varphi_r^*(x),
]

belongs to the class (W^{(r)}H^{(\alpha)}) and satisfies the inequalities (3).

From Theorem 1 it follows:

Theorem 2. In order that the linear method of approximation
(U_n(f;x;\lambda)), as (n\to\infty), should realize, for every function
(f(x)\in W^{(r)}H^{(\alpha)}), approximation of the same order as the best approximation, it is necessary that

[
\sum_{k=1}^{n}\frac{\mu_k^{(n)}}{n-k+1}
=
\frac{\ln n}{n^r}
+
O\left(\frac{1}{n^r}\right).
]

Theorem 2 supplements Theorem 4 of work ({}^{1}).

I consider it my duty to express my deep gratitude to A. F. Timan for his help in carrying out the present work.

Dnepropetrovsk
Metallurgical Institute

Received
15 I 1958

REFERENCES

({}^{1}) G. Ya. Doronin, DAN, 69, No. 4 (1949).
({}^{2}) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 10, 393 (1946).
({}^{3}) A. F. Timan, Izv. AN SSSR, ser. matem., 17, No. 2 (1953).