UDC 517.512.6

MATHEMATICS

V. G. VERDIEV

ON THE LOCATION OF THE POINTS OF MAXIMAL DEVIATION IN CHEBYSHEV APPROXIMATION OF CONTINUOUS FUNCTIONS

(Presented by Academician S. N. Bernstein on 6 XI 1968)

Let \(f(t)\in C_{2\pi}\); let \(T_n(t;f)\) be the trigonometric polynomial of order \(\le n\) least deviating from \(f(t)\) in the metric of the space \(C_{2\pi}\);

\[ E_n^*(f)=\max |f(t)-T_n(t;f)|\quad (0\le t\le 2\pi); \]

let \(M_n^*(f)\) be the set of points \(t\in [0,2\pi)\) where

\[ E_n^*(f)=|f(t)-T_n(t;f)|. \]

Put

\[ M^*(f)=\bigcup_{n=0}^{\infty} M_n^*(f). \]

For \(f(x)\in C[-1,1]\), \(E_n(f)\) and \(M(f)\) are defined analogously.

In paper \((^1)\) M. I. Kadets formulated a theorem which is a generalization of Theorem II \((^2)\), p. 88, of S. N. Bernstein, and from which it follows that the set \(M(f)\) of any function \(f(x)\in C[-1,1]\) is everywhere dense on \([-1,1]\). The aim of the present note is to prove that the set \(M^*(f)\) of any function \(f(t)\in C_{2\pi}\) is everywhere dense on the interval \([0,2\pi]\).

As noted in \((^1)\), the series

\[ \sum_{n=0}^{\infty}\frac{E_n^*(f)-E_{n+1}^*(f)}{E_n^*(f)+E_{n+1}^*(f)} \]

diverges, whence it follows that

\[ \varlimsup_{n\to\infty}\left(\frac{E_n^*(f)-E_{n+1}^*(f)}{E_n^*(f)+E_{n+1}^*(f)}\right)^{1/n}=1. \]

Lemma 1. If \(f(t)\in C_{2\pi}\) and \(E_p^*(f)=E_{r-1}^*(f)>E_r^*(f)\) \((p<r-1)\), then the polynomial \(T_r^*(t)=T_p(t;f)-T_r(t;f)\) has \(2r\) simple zeros \(z_1<z_2<\cdots<z_{2r}\in(0,2\pi)\); the points \(\{u_i\}_{i=1}^{2r}\) of maximal deviation from zero of the difference \(f(t)-T_p(t;f)\) alternate with the zeros \(\{z_i\}\) of the polynomial \(T_r^*(t)\).

Since \(E_p^*(f)=E_{r-1}^*(f)\), we have \(T_p(t;f)=T_{r-1}(t;f)\).

For the proof of Lemma 1 it suffices to note that

\[ T_p(t;f)-T_r(t;f)=\bigl[T_{r-1}(t;f)-f(t)\bigr]-\bigl[T_r(t;f)-f(t)\bigr] \]

and that \(E_{r-1}^*(f)>E_r^*(f)\).

Denote by \(W_n(h)\) the class of trigonometric polynomials \(T(t)\) of order \(n\) having the following properties:

1) the polynomial \(T(t)\) has, in \((0,2\pi)\), \(2n\) simple zeros

\[ z_1<z_2<\cdots<z_{2n}; \]

2)

\[ \min_{0\le i\le 2n}\|T\|_{[z_i,z_{i+1}]}\ge h\|T\|_{[0,2\pi]}\quad (z_0=0,\ z_{2n+1}=2\pi;\ 0<h\le 1). \]

Lemma 2. Let \(x_1,x_2,\ldots,x_{2n}\in(0,2\pi)\) be the zeros of a polynomial \(T(t)\in W_n(h)\); let \(t_1,t_2,\ldots,t_{2n}\in[0,2\pi]\) be the zeros of its derivative \(T'(t)\); let \(y_1,y_2,\ldots,y_{2n}\in(0,2\pi)\) be the zeros of a polynomial \(Q(t)\in W_n(h)\); and let \(q_1,q_2,\ldots,q_{2n}\in[0,2\pi]\) be the zeros of its derivative \(Q'(t)\). If \(T(t_i)=Q(q_i)\) for \(i=0,1,\ldots,j-1,j+1,\ldots,2n\) and \(T(t_j)>Q(q_j)>0\) \((T(t_j)<Q(q_j)<0)\), then

\[ 0\le t_1<q_1<t_2<q_2<\cdots<t_{j-1}<q_{j-1}<q_{j+1}<t_{j+1}<q_{j+2}< \]

\[ <t_{j+2}<\cdots<q_{2n}<t_{2n}<2\pi; \tag{1} \]

\[ 0<x_1<y_1<x_2<y_2<\ldots<x_j<y_j<y_{j+1}<x_{j+1}<\ldots \]

\[ \ldots<y_{2n}<x_{2n}<2\pi; \tag{2} \]

\[ y_1-x_1<y_2-x_2<\ldots<y_j-x_j; \]

\[ x_{j+1}-y_{j+1}>x_{j+2}-y_{j+2}>\ldots>x_{2n}-y_{2n}. \tag{3} \]

As in the proof of Lemma 1 in (3), here one can also indicate the process of passing from the polynomial \(T(t)\) to the polynomial \(Q(t)\), from which (1) and (2) will follow.

Let us prove assertion (3). Suppose that, for some fixed \(i\) \((1<i\leqslant j)\), \(y_i-x_i\leqslant y_{i-1}-x_{i-1}\). By the condition of Lemma 2, on the interval \([0,2\pi)\) the trigonometric polynomial
\[ T(t)-Q(t+(y_{i-1}-x_{i-1}))=F(t) \]
of order \(n\) must have at least \(i\) zeros in the half-interval \([0,x_i)\), and at least \(2n-i\) zeros in the interval \((x_i,2\pi)\); moreover, \(x_i\) is a zero of \(F(t)\). Since \(F(t)\) can have \(\leqslant 2n\) zeros in the interval \([0,2\pi)\), our supposition is false.

Denote by \(\tau(t;i)\) \((i=0,1,\ldots,2n)\) the trigonometric polynomial \(\in W_n(h)\) such that
\[ \tau(\tau_l^i;i)= \begin{cases} (-1)^{2n-l} & \text{for } l=0,1,\ldots,i-1,\\ (-1)^{2n-l} & \text{for } l=i+1,\ldots,2n, \end{cases} \]
and \(\tau'(\tau_j^i;i)=0\) for \(j=0,1,\ldots,2n\), where \(\tau_j^i\in[0,2\pi)\). The existence and uniqueness of such polynomials are proved in (4). The polynomial \(\tau(t;n)\) has the form (see, for example, (5, 6))
\[ \tau(t;n)=h\cos 2n\arccos\left(\frac{\sin(t/2)}{\sin(\omega_n/2)}\right), \]
where \(\omega_n\) is chosen so that \(\tau(\pi;n)=1\), i.e., from the condition
\[ \tg^{2n}\omega_n/4+\ctg^{2n}\omega_n/4=2/h. \tag{4} \]

For the polynomials \(\{\tau(t;i)\}_{i=0}^{2n}\) the equality
\[ \tau(t;i)=(-1)^{\,n-i}\tau(t+\alpha_i;n), \]
holds, where \(\alpha_i\) is a constant depending on \(i\).

Put
\[ d_n(T;h)=\max |z_i^T-z_{i+1}^T|,\quad 0\leqslant i\leqslant 2n;\qquad D_n(h)=\sup_{T(t)\in W_n(h)} d_n(T;h), \]
where \(z_1^T,z_2^T,\ldots,z_{2n}^T\in(0,2\pi)\) are the zeros of the polynomial \(T(t)\in W_n(h)\), \(z_0^T=0\), \(z_{2n+1}^T=2\pi\).

Theorem 1. The upper bound \(D_n(h)\) is attained on the polynomials \(\tau(t;i)\) and is equal to
\[ D_n(h)=2\left(\pi-2\arcsin\left(\frac{\sin\omega_n}{2\cos(4\pi-1)\pi/8n^2}\right)\right)=O(1/\sqrt[n]{h}). \]

We shall show that \(D_n(h)\) cannot be attained on a polynomial \(T(t)\in W_n(h)\) for which \(\|T\|_{C_{2\pi}}<1\). Suppose, to the contrary, that
\[ D_n(h)=|z_i^T-z_{i+1}^T|, \]
where \(z_i^T\) and \(z_{i+1}^T\) are neighboring zeros of the polynomial \(T(t)\). Let
\[ t_0<t_1<\ldots<t_{2n}\in[0,2\pi) \]
be the zeros of the derivative \(T'(t)\) of the polynomial \(T(t)\), and suppose, for definiteness, that \(T(t_i)>0\), where \(z_i^T<t_i<z_{i+1}^T\).

Consider a polynomial \(S(t)\in W_n(h)\) such that \(S(s_j)=T(t_j)\) for
\[ j=0,1,\ldots,i-1,i+1,\ldots,2n; \]
\(S(s_i)=1\), where the \(s_j\) are the zeros of the derivative \(S'(t)\) of the polynomial \(S(t)\), lying in the interval \([0,2\pi)\). Let \(z_i^S\) and \(z_{i+1}^S\in(0,2\pi)\) be neighboring zeros of the polynomial \(S(t)\). Then from Lemma 2 we have
\[ z_i^S<z_i^T \quad\text{and}\quad z_{i+1}^T>z_{i+1}^S, \]
or
\[ D_n(h)=|z_i^T-z_{i+1}^T|<|z_i^S-z_{i+1}^S|. \]
Show that \(D_n(h)\) also cannot be attained on a polynomial \(Q(t)\ne\tau(t;i)\) from \(W_n(h)\). Suppose, to the contrary, that \(D_n(h)\) is attained on a polynomial \(Q(t)\ne\tau(t;i)\) for the zeros \(z_i^q\) and \(z_{i+1}^q\), i.e.,
\[ D_n(h)=|z_i^q-z_{i+1}^q|. \]
If \(Q(t)\ne\tau(t;i)\), then at least one of the inequalities
\[ Q(q_l)\geqslant \tau(\tau_l^i;i)\qquad (l=0,1,\ldots,2n), \]
holds, where \(q_l\in[0,2\pi)\) and \(\tau_l^i\in[0,2\pi)\) are, respectively, points of extremum of the polynomials \(Q(t)\) and \(\tau(t;i)\). For definiteness, let
\[ Q(q_j)>\tau(\tau_j^i;i)>0\qquad (j\ne i). \]
Then from Lemma 2, by virtue of inequalities (3), we have
\[ z_i^\tau<z_i^q<z_{i+1}^q<z_{i+1}^\tau \]
or
\[ |z_i^q-z_{i+1}^q|<|z_i^\tau-z_{i+1}^\tau|, \]
where \(z_i^\tau\) and \(z_{i+1}^\tau\) are the \(i\)-th and \((i+1)\)-st zeros of the polynomial \(\tau(t;i)\) on the interval \([0,2\pi]\). The quantity \(D_n(h)\) is determined by direct calculations. Theorem 1 is proved.

Let the sequence of numbers $\{h_n\}_{n=1}^{\infty}$ $(0<h_n<1,\ n=1,2,\ldots)$ be such that

\[ \lim_{n\to\infty}\sqrt[n]{h_n}=1; \tag{5} \]

$\{W_n(h_n)\}_{n=1}^{\infty}$ is the sequence of classes of polynomials corresponding to $\{h_n\}_{n=1}^{\infty}$; $\{T_n(t)\}_{n=1}^{\infty}=\mathcal T$ is a sequence of polynomials such that $T_n(t)\in W_n(h_n)$ for any $n=1,2,\ldots$. Denote by $Z_n$ the set of zeros of the polynomial $T_n(t)\in W_n(h_n)$ situated in the interval $[0,2\pi)$, and put $Z=\bigcup_{n=1}^{\infty} Z_n$. We shall call $Z$ the set of zeros of $\mathcal T$.

Theorem 2. The set $Z$ of zeros of the sequence $\mathcal T$ is everywhere dense on the interval $[0,2\pi]$.

We have $\lvert z^T_{i,n}-z^T_{i+1,n}\rvert\leq D_n(h_n)$. By (5), it follows from (4) that $\lim \omega_n=\pi$ as $n\to\infty$. Hence, and from Theorem 1, it follows that $\lim D_n(h_n)=0$ as $n\to\infty$. Theorem 2 is proved.

Theorem 3. The set $M^*(f)$ of functions $f(t)\in C_{2\pi}$ is everywhere dense on the interval $[0,2\pi]$.

The polynomial
\[ T^*_{n_{k+1}}(t)=T_{n_k}(t;f)-T_{n_{k+1}}(t;f)\in W_{n_{k+1}}(h_{n_{k+1}}), \]
where
\[ h_{n_{k+1}}=\{E^*_{n_k}(f)-E^*_{n_{k+1}}(f)\}/\{E^*_{n_k}(f)+E^*_{n_{k+1}}(f)\} \quad (k=0,1,2,\ldots). \]
Since $f(t)\in C_{2\pi}$, we have
\[ \lim \sqrt[n_k]{h_{n_k}}=1 \]
as $k\to\infty$. Therefore, by Theorem 2, the set $Z$ of zeros of the sequence $\{T^*_{n_{k+1}}(t)\}_{k=0}^{\infty}$ is everywhere dense on $[0,2\pi]$. By virtue of Lemma 1, the zeros of $T^*_{n_{k+1}}(t)$ alternate with the points of maximum deviation from zero of the difference $f(t)-T_{n_k}(t,f)$ for every $k=0,1,2,\ldots$. Theorem 3 is proved.

The author expresses his gratitude to V. S. Videnskii for his attention to this work.

Leningrad Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich

Received
21 IX 1968

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