Full Text
UDC 517.512.6
MATHEMATICS
D. L. BERMAN
INVESTIGATION OF INTERPOLATION PROCESSES CONSTRUCTED FOR AN EXTENDED SYSTEM OF P. L. CHEBYSHEV NODES
(Presented by Academician S. N. Bernstein on 28 XI 1966)
1°. Let a matrix of numbers be given
\[ -1 \le x_n^{(n)} < x_{n-1}^{(n)} < \cdots < x_1^{(n)} \le 1,\qquad n=1,2,\ldots \tag{m} \]
The function \(\theta=\arccos x,\ -1\le x\le 1\), maps the matrix \((m)\) one-to-one onto the matrix
\[ 0\le \theta_1^{(n)}<\theta_2^{(n)}<\cdots<\theta_n^{(n)}\le \pi . \]
We shall call the point \(\theta_k^{(n)}\) the image of the point \(x_k^{(n)}\) on the unit semicircle. It has been known for a comparatively long time that the law of distribution of the images \(\{\theta_l^{(n)}\}\) plays an important role in the theory of interpolation. Thus, for example, in order that the Lagrange interpolation process
\[ L_n(f,x)=\sum_{k=1}^{n} f\bigl(x_k^{(n)}\bigr) l_k^{(n)}(x),\qquad l_k=l_k^{(n)}(x)=\frac{\omega_n(x)}{(x-x_k^{(n)})\omega_n'(x_k^{(n)})}, \]
\[ \omega_n(x)=\prod_{j=1}^{n} (x-x_j^{(n)}), \]
converge uniformly for every function \(f(x)\) continuous on \([-1,1]\), it is necessary that the inequalities
\[
C_1/n \le \theta_{k+1}^{(n)}-\theta_k^{(n)} \le C_2/n,\qquad
k=0,1,2,\ldots,n;
\]
\[
n=1,2,\ldots;\quad \theta_0=0;\quad \theta_{n+1}=\pi,
\tag{1}
\]
hold, where the constants \(C_i>0,\ i=1,2\), do not depend on \(n\) \((^1)\). An analogous result is known \((^1)\) for the Hermite–Fejér interpolation process.
If the matrix \((m)\) is such that inequalities (1) are satisfied, then we say that the nodes \(m\) are quasi-uniformly distributed.
According to the classical result of S. N. Bernstein–G. Faber, there is no such matrix of nodes \((m)\) for which, for every \(f\in C\), the relation*
\[ L_n(f,x)\to f(x),\qquad n\to\infty \tag{2} \]
holds uniformly on \([-1,1]\).
Therefore S. N. Bernstein \((^2,\ \text{p. }500)\) posed the problem of constructing such an interpolation process \(\{A_n(f,x)\}_{n=1}^{\infty}\) which converges uniformly for every \(f\in C\) and at the same time has the property that the ratio \(\sigma_n\) of the degree of the polynomial \(A_n\) to the number of its nodes can be made arbitrarily close to one. This problem was solved by S. N. Bernstein himself \((^3)\), who constructed the polynomials \(A_n\) in the following way.
For a given fixed natural number \(p\), put
\[ A_n(f,x)=\sum_{k=1}^{n} a_k^{(n)}(f)l_k(x), \tag{3} \]
where
\[ a_k^{(n)}=f\bigl(x_k^{(n)}\bigr),\qquad k\not\equiv 0 \pmod{2p}; \]
\[ a_{2pt}^{(n)}=\sum_{j=1}^{p} f\bigl(x_{2p(t-1)+2j-1}^{(n)}\bigr) -\sum_{j=1}^{p-1} f\bigl(x_{2p(t-1)+2j}^{(n)}\bigr). \]
\[ \text{* } C \text{ is the set of all functions continuous on }[-1,1]. \]
It is not hard to see that, for sufficiently large \(p\), \(\sigma_n\) is arbitrarily close to unity. S. N. Bernstein \((^3)\) proved that, for the nodes of P. L. Chebyshev
\[ x_k^{(n)}=\cos \frac{2k-1}{2n}\pi,\qquad k=1,2,\ldots,n;\ n=1,2,\ldots, \tag{4} \]
for every \(f\in C\) the relation
\[ A_n(f,x)\to f(x),\qquad n\to\infty,\qquad -1\le x\le 1 \tag{5} \]
holds uniformly. The polynomials \(A_n\) of S. N. Bernstein are remarkable also because there exists \((^4)\) a very broad class of node matrices \((m)\), including the nodes (4), for which, for every \(f\in C\), the relation (5) holds uniformly. A natural question arises: what must the nodes be in order that, for every \(f\in C\), the interpolation process of S. N. Bernstein converge uniformly?
In \((^5,^6)\) it was proved that, for this, it is necessary that the nodes be quasi-uniformly distributed.* The question of whether this condition is sufficient remained open. Theorem 1 gives an answer to this question.
2°. By an extended system of Chebyshev nodes we shall mean the node matrix
\[ x_0^{(n+2)}=1,\quad x_{n+1}^{(n+2)}=-1,\quad x_k^{(n+2)}=\cos \frac{2k-1}{2n}\pi, \]
\[ k=1,2,\ldots,n;\ n=1,2,\ldots . \tag{6} \]
It is obvious that these nodes are quasi-uniformly distributed.
Theorem 1. The interpolation process (3) of S. N. Bernstein, constructed at the nodes (6) for the function \(\mu(x)=(x+|x|)/2\), diverges at the point \(x=0\).
We outline the proof. We shall assume that the polynomial \(A_n\) is constructed with \(p=1\). Then it is not hard to see that
\[ A_n(\mu,0)=x_0(l_0+l_1)+\cdots+x_{2p}(l_{2p}+2l_{2p+1}),\qquad n=4p. \]
Hence, after simple calculations, we obtain
\[ A_n(\mu,0)=\frac12+\frac{1}{n\cos \pi/2n}-\gamma_n, \]
\[ \gamma_n=\frac1n\sum_{k=1}^{2p-1}{}' \frac1{x_k} \left( \frac{x_{k-1}}{\sqrt{1-x_k^2}}-\frac{x_k}{\sqrt{1-x_{k-1}^2}} \right). \]
The prime means that the summation is taken over odd \(k\).
With the help of straightforward computations we obtain
\[ A_n(\mu,0)>\frac12+\frac{1}{n\cos \pi/2n}-\frac{682}{495\pi}-\tau_n,\qquad \tau_n\to 0,\quad n\to\infty. \]
It follows that
\[ \lim_{n\to\infty} A_n(\mu,0)>1/11\pi . \]
Consequently, the process diverges at the point \(x=0\).
3°. Denote by \(\mathfrak M_\tau\), \(\tau\ge 1\), the set of all node matrices \((m)\) for which the quantity
\[ \lambda_n^{(\tau)}=\lambda_n^{(\tau)}(m)= \max_{-1\le x\le 1}\left(\sum_{j=1}^n |l_j(x)|^\tau\right)^{1/\tau} \]
is bounded, i.e. \(\lambda_n^{(\tau)}\le C\), where \(C\) depends only on \((m)\) and \(\tau\). Among the numbers \(\lambda_n^{(\tau)}\), the quantity \(\lambda_n^{(1)}\), which is the Lebesgue constant of the Lagrange interpolation process, is of special interest. The quantity \(\lambda_n^{(\tau)}\), as a rule, is difficult to estimate. It is easy to see that
\[ \lambda_n^{(1)}\le n^{1-1/\tau}\lambda_n^{(\tau)}. \tag{7} \]
Therefore an upper estimate of the quantities \(\lambda_n^{(\tau)}\) is of interest.
Theorem 2. In the case of the nodes (6) the inequality
\[ \lambda_n^{(2)}\le \sqrt{7}. \tag{8} \]
\[ \text{* I.e., that the inequalities (1) be satisfied.} \]
We outline the proof. For the nodes (6)
\[ \begin{aligned} (\lambda_m^{(2)})^2 &=\frac{(x^2+1)T_n^2(x)}{2} +\frac{T_n^2(x)}{n^2}\sum_{k=1}^{n}\frac{1-x^2}{(x-x_k)^2} -\frac{T_n^2(x)}{n^2}\left\{\sum_{k=1}^{n}\frac{2x}{x-x_k}\right\} \\ &\quad+\frac{(1+x)^2T_n^2(x)}{2n^2}\sum_{k=1}^{n}\frac{1}{1-x_k} +\frac{(1-x)^2T_n^2(x)}{2n^2}\sum_{k=1}^{n}\frac{1}{1+x_k} =S_0+S_1-S_2+S_3+S_4 . \end{aligned} \tag{9} \]
Since
\[ \frac{T_n'(x)}{T_n(x)}=\sum_{k=1}^{n}\frac{1}{x-x_k}; \tag{10} \]
\[ \frac{(T_n'(x))^2-T_n''(x)T_n'(x)}{T_n^2(x)} =\sum_{k=1}^{n}\frac{1}{(x-x_k)^2}, \]
we have
\[ S_1=1-\frac{\sin 2n\theta\cos\theta}{2n\sin\theta}. \tag{11} \]
Consequently, \(0\leq S_1\leq 1\).
We pass to the estimate of \(S_2\). From (9) it is seen that
\[ S_2=\frac{2xT_n(x)}{n}\sum_{k=1}^{n} l_k(x)\frac{(-1)^{k-1}}{\sqrt{1-x_k^2}}, \]
where \(\{l_k(x)\}_{k=1}^{n}\) are the fundamental polynomials of the Chebyshev nodes. Since
\[ \tag{7} \sum_{k=1}^{n} l_k^2(x)\leq 2,\qquad -1\leq x\leq 1, \]
it follows that
\[ |S_2|\leq \frac{2x|T_n(x)|}{n}\sqrt{2}\left(\sum_{k=1}^{n}\frac{1}{1-x_k^2}\right)^{1/2}. \tag{12} \]
In view of the fact that (8)*
\[ \sum_{k=1}^{n}\frac{1}{\cos^2\theta_k/2} =\sum_{k=1}^{n}\frac{1}{\sin^2\theta_k/2} =2n^2,\qquad \theta_k=\frac{2k-1}{2n}\pi, \tag{13} \]
from (12) we obtain
\[ |S_2|\leq 2x|T_n(x)|\sqrt{2}\leq 2\sqrt{2}. \tag{14} \]
The identities (13) lead to the equality
\[ S_3+S_4=(1+x^2)T_n^2(x). \]
Thus,
\[ S_0+S_3+S_4=\frac{3(1+x^2)}{2}T_n^2(x)\leq 3,\qquad -1\leq x\leq 1. \tag{15} \]
From (11), (14), (15) follows (8).
Corollary 1. For the nodes (6),
\[ \lambda_n^{(1)}\leq \sqrt{7n}. \]
This assertion follows directly from Theorem 2 and inequality (7).
Corollary 2. If the best approximation \(E_n(f)\) of the function \(f\) by a polynomial of degree \(n\) satisfies the inequality
\[ E_n(f)\leq C/n^\alpha,\qquad \alpha>1/2, \]
then the Lagrange interpolation process constructed for the function \(f\) at the nodes (6) converges uniformly to \(f\) on \([-1,1]\).
* These equalities follow very simply from identity (10). In (8) they are proved differently.
Indeed, the inequality is known ((2), p. 258)
\[ |f(x)-L_n(f,x)| \leq (1+\lambda_n^{(1)})F_n(f). \]
Therefore Corollary 2 follows directly from Corollary 1.
From Corollary 2, in particular, it follows that the Lagrange interpolation process constructed at the nodes (6) for the function \(f(x)=|x|\) converges uniformly on \([-1,1]\), while the Hermite–Fejér interpolation process constructed for the same function and at the same nodes diverges at \(x=0\) (9).
4°. Corollary 2 can be strengthened. To this end we first establish a theorem:
Theorem 3. Let the fundamental Lagrange polynomials \(\{l_j(x)\}\) be constructed at the nodes (6), and
\[ M_n(x)=\sum_{j=0}^{n+1}|l_j(x)|. \]
Then the estimate
\[ M_n(x) \leq \frac{4}{\pi}|T_n(x)|\ln n+20,\qquad n\geq 10,\quad -1\leq x\leq 1 \tag{16} \]
is valid.
We outline the proof. By virtue of the symmetry of the nodes, we may assume that \(0\leq x<1\). Since \(|l_0(x)|+|l_{n+1}(x)|\leq 1\), \(-1\leq x\leq 1\), we shall estimate the function
\[ \psi(x)=\sum_{j=1}^{n}|l_j(x)|. \]
If \(x\) is a node, then (16) is obvious. Therefore let us suppose that \(x_{p+1}<x<x_p\). We have
\[ \psi(x)=\sum_{j=1}^{p-2}|l_j(x)|+ \sum_{j=p-1}^{p+2}|l_j(x)|+ \sum_{j=p+3}^{n}|l_j(x)|\equiv S_1+S_2+S_3. \]
By Theorem 2,
\[ |S_2|\leq 4\sqrt{7}. \tag{17} \]
Let
\[ r_1(t)=\frac{1}{(\cos t-\cos\theta)\sin t},\qquad 0\leq t<\theta; \]
\[ r_2(t)=\frac{1}{(\cos\theta-\cos t)\sin t},\qquad \theta<t<\pi; \]
\[ \theta^{(+)}=\arccos\frac{x+\sqrt{8+x^2}}{4},\qquad \theta^{(-)}=\arccos\frac{x-\sqrt{8+x^2}}{4},\qquad 0\leq x<1. \]
It is not difficult to verify that \(r_1(t)\) decreases on the interval \((0,\theta^{(+)})\) and increases on \((\theta^{(+)},\theta)\); \(r_2(t)\) decreases on \((\theta,\theta^{(-)})\) and increases on \((\theta^{(-)},\pi)\). With the aid of these properties of the functions \(r_i(t)\), \(i=1,2\), after some calculations we obtain
\[ S_1+S_3\leq \frac{|T_n(x)|}{\pi} \left( \ln\operatorname{ctg}\frac{\theta-\theta_{p-1}}{2}\, \operatorname{ctg}\frac{\theta_{p+2}-\theta}{2}\, \cosec\theta_1\,\cosec\theta_n + \right. \]
\[ \left. +\cos\theta\ln\operatorname{tg}\frac{\theta_{p-1}}{2}\, \operatorname{tg}\frac{\theta_{p+2}}{2} \right) +2\sqrt{7}\qquad (x=\cos\theta). \]
Hence, together with (17), (16) follows.
Theorem 4. The Lagrange interpolation process \(\{L_n(f)\}\), constructed at the nodes (6) for a function \(f\) from the Dini–Lipschitz class \((\lim \omega_f(\delta)\ln\delta\to 0)\), satisfies relation (2) uniformly on \([-1,1]\).
Leningrad Institute of Soviet Trade
named after Fr. Engels
Received
20 X 1966
References
- P. Erdös, P. Turán, Ann. Math., 39, 703 (1938).
- S. N. Bernstein, Collected Works, 1, Publishing House of the Academy of Sciences of the USSR, 1952.
- S. N. Bernstein, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954, p. 130.
- D. L. Berman, DAN, 60, No. 3 (1948).
- D. L. Berman, DAN, 119, No. 6 (1958).
- D. L. Berman, Izv. vyssh. uchebn. zaved., matem., No. 1 (1957).
- L. Fejér, Math. Ann., 106, 1 (1932).
- M. Riesz, C. R., 158, 1152 (1914).
- D. L. Berman, DAN, 163, No. 3 (1965).