Abstract
Full Text
UDC 517.512.6
MATHEMATICS
D. L. BERMAN
AN INVESTIGATION OF THE HERMITE–FEJÉR INTERPOLATION PROCESS
(Presented by Academician S. N. Bernstein on November 6, 1968)
1°. Let a matrix of numbers be given
\[
-1 \le x_n^{(n)} < x_{n-1}^{(n)} < \cdots < x_1^{(n)} \le 1,\qquad n=1,2,\ldots
\tag{m}
\]
Denote by \(C\) the set of all functions \(f(x)\) continuous on \([-1,1]\). By \(L_n(f,x)\) we shall denote the Lagrange interpolation polynomial of degree \((n-1)\), constructed for the function \(f(x)\) and the \(n\)-th row of the matrix \((m)\).
According to the classical theorem of S. N. Bernstein—G. Fejér, there is no matrix of nodes \((m)\) for which, for every \(f\in C\), the relation \(L_n(f,x)\to f(x)\), \(n\to\infty\), holds uniformly on \([-1,1]\).
In this connection the theorem of L. Fejér is of interest; it is formulated as follows. Let the matrix \((m)\) be composed of the roots of the polynomials of P. L. Chebyshev
\(T_n(x)=\cos n\arccos x\), i.e.
\[
x_k^{(n)}=\cos \frac{2k-1}{2n}\pi,\qquad k=1,2,\ldots,n;\ n=1,2,\ldots,
\tag{1}
\]
and let \(H_n(f)\equiv H_n(f,x)\) be the polynomial of degree \((2n-1)\), constructed for \(f\in C\) and the nodes (1), and uniquely determined by the conditions
\(H_n(f,x_k^{(n)})=f(x_k^{(n)})\),
\(H_n'(f,x_k^{(n)})=0\), \(k=1,2,\ldots,n\). Then for every \(f\in C\) the relation \(H_n(f,x)\to f(x)\), \(n\to\infty\), holds uniformly on \([-1,1]\).
The interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\) is usually called the Hermite–Fejér interpolation process. It was the subject of important investigations by L. Fejér and his students.
2°. In \((^1)\) it was found that if the matrix \((m)\) is composed of the numbers
\[
x_{n+1}^{(n+2)}=-1;\qquad
x_k^{(n+2)}=\cos \frac{2k-1}{2n}\pi,\quad k=1,2,\ldots,n;
\]
\[
x_0^{(n+2)}=1,\quad n=1,2,\ldots,
\tag{2}
\]
then the interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=|x|\), diverges at the point \(x=0\). Since the matrix (2) is obtained from the matrix of Chebyshev nodes by adding the points \(x=\pm1\) as nodes, in connection with Fejér’s theorem this result seems unexpected. Quite naturally there arose the question whether there exists such a continuous function on \([-1,1]\) for which the process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed at the nodes (2), diverges at all points of the interval \((-1,1)\). The answer to this question is given by Theorem 1.
Theorem 1. The interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=1-x^2\) at the nodes (2), diverges at every point of the interval \((-1,1)\).
We outline the proof. For the nodes (2),
\[ \begin{aligned} H_{n+2}(f,x)={}&\frac{f(1)}{2}\,[1-(2n^2+1)(x-1)](x+1)T_n'(x) \\ &-\frac{f(-1)}{2}\,[1+(2n^2+1)(x+1)](x-1)T_n'(x) \\ &+\sum_{j=1}^{n} f(x_j)\left(1+\frac{3x_j(x-x_j)}{1-x_j^2}\right) \frac{4x_j(x^2-1)^2T_n^2(x)} {(x^2-x_j^2)(x_j^2-1)[T_n'(x_j)]^2}. \end{aligned} \]
Therefore, for \(f(x)=1-x^2,\ n=4p\),
\[ H_{n+2}(f,x)= \frac{2(x^2-1)T_n^2(x)}{n^2} \sum_{j=1}^{2p} \frac{4x_j^4-(2x^2+1)x_j^2-x^2} {(1-x_j^2)(x^2-x_j^2)^2}. \]
Decomposing the fraction under the \(\sum\) into partial fractions and taking into account the symmetry of the nodes, we obtain
\[ H_{n+2}(f,x)= \frac{(1-x^2)T_n^2(x)}{n^2} \left( \frac{3}{x^2-1}\sum_{j=1}^{n}\frac{1}{1-x_j} + \sum_{j=1}^{n}\frac{1}{(x-x_j)^2} + \frac{3x}{1-x^2}\sum_{j=1}^{n}\frac{1}{x-x_j} \right). \]
We now use the identities:
\[ \sum_{k=1}^{n}\frac{1}{\sin^2\theta_k/2}=2n^2, \qquad \theta_k=\frac{(2k-1)\pi}{2n}, \tag{3} \]
\[ \frac{T_n'(x)}{T_n(x)}=\sum_{j=1}^{n}\frac{1}{x-x_j}, \qquad \frac{T_n^2(x)(1-x^2)}{n^2} \sum_{j=1}^{n}\frac{1}{(x-x_j)^2} = 1-\frac{\sin 2n\theta\cos\theta}{2n\sin\theta}, \qquad x=\cos\theta. \]
Consequently,
\[ H_{n+2}(f,x)=1-3\cos^2 n\theta+\frac{\sin 2n\theta\cos\theta}{n\sin\theta}, \qquad x=\cos\theta. \tag{4} \]
Since the process is symmetric, we may assume that \(0\le x<1\). Suppose that at \(\bar{x}\in[0,1)\) the process (4) converges. Then
\[ \lim_{n\to\infty}\sin^2 n\theta=1-\frac{\cos^2\theta}{3}, \qquad \cos\theta=\bar{x}, \]
and this contradicts the following lemma.
Lemma. For any \(\theta\) from \([0,\pi/2]\) one can find a sequence of natural numbers \(n_1<n_2<\cdots\), \(n_k\to\infty,\ k\to\infty\), such that the equality
\[
\lim_{k\to\infty}\sin^2 n_k\theta=0
\]
holds.
Theorem 2. The interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=x^2\) at the nodes (2), diverges at every point of the interval \((-1,1)\).
The proof follows from Theorem 1 and the equality
\[ H_{n+2}(1-z^2,x)=1-H_{n+2}(z^2,x). \]
In connection with Theorem 2 it is of interest to investigate the convergence of the process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed at the nodes (2), for the function \(f(x)=x\).
Theorem 3. The process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=x\) at the nodes (2), diverges at all points of \((-1,1)\) except \(x=0\).
We outline the proof. For \(f(x)=x\) the identity holds
\[ R_n(x)\equiv x-H_{n+2}(f,x)= \sum_{k=0}^{n+1}(x-x_k)L_k^2(x), \]
where \(\{L_k(x)\}\) are the fundamental Lagrange polynomials of the nodes (2). After elementary transformations we obtain that
\[ R_n(x)=\frac{\sin 2n\theta\sin\theta}{2n} -\frac{3\cos\theta\sin^2\theta}{2}\cos^2 n\theta. \tag{5} \]
Hence, with the aid of the lemma formulated earlier, it follows that the process diverges at every point \(x \ne 0\) of \((-1,1)\). The convergence of the process for \(x=0\) is obvious, since for \(x=0\) the right-hand side of (5) is equal to zero.
\(3^\circ\). The results of item \(2^\circ\) show that extending the Chebyshev nodes by adding the points \(x=\pm 1\) as nodes substantially worsens the behavior of the Hermite–Fejér interpolation process. The question arises: does this situation occur for every system of nodes? From the following theorems it is clear that this is not always so.
Theorem 4. The interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=x^2\) at the nodes
\[ x_k^{(n+2)}=\cos k\pi/(n+1),\qquad k=0,1,\ldots,(n+1);\quad n=1,2,\ldots, \tag{6} \]
converges uniformly on \([-1,1]\). Moreover,
\[ |H_{n+2}(f,x)-x^2|\leq 4/(n+1). \tag{7} \]
We outline the proof. For the nodes (6),
\[ \begin{aligned} H_{n+2}(f,x)=\frac{U_n^2(x)}{(n+1)^2}\Bigg\{& \frac{f(1)}{4}\left[1-\left(1+\frac{2}{3}n(n+2)\right)(x-1)\right](x+1)^2 \\ &+\frac{f(-1)}{4}\left[1+\left(1+\frac{2}{3}n(n+2)\right)(x+1)\right](x-1)^2 \\ &+\sum_{j=1}^{n} f(x_j)\left(1-\frac{x_j}{x_j^2-1}(x-x_j)\right) \left(\frac{x^2-1}{x-x_j}\right)^2 \Bigg\}, \end{aligned} \]
\[ U_n(x)=\sin(n+1)\theta/\sin\theta,\qquad x=\cos\theta. \]
It is obvious that it suffices to prove the theorem for \(f(x)=1-x^2\). In this case
\[ H_{n+2}(f,x)= \frac{(1-x^2)^2U_n^2(x)}{(n+1)^2} \sum_{j=1}^{n} \frac{1+xx_j-2x_j^2}{(x-x_j)^2}. \tag{8} \]
Hence, after elementary transformations using the differential equation for the Chebyshev polynomials of the second kind, we obtain
\[ H_{n+2}(f,x)=\frac{1}{(n+1)^2} \left[\sin^2 n\theta-2n\sin n\theta\cos(n+1)\theta\sin\theta+n^2\sin^2\theta\right], \]
\[ x=\cos\theta. \tag{9} \]
Estimate (7) follows immediately from (9).
Remark. From (8) and (9) follows the identity
\[ \sin^2(n+1)\theta \sum_{j=1}^{n} \frac{1+\cos\theta_j\cos\theta-2\cos^2\theta_j} {(\cos\theta-\cos\theta_j)^2} = \]
\[ = n^2-2n\frac{\sin n\theta}{\sin\theta}\cos(n+1)\theta +\left(\frac{\sin n\theta}{\sin\theta}\right)^2, \qquad \theta_j=\cos\frac{j\pi}{n+1}. \]
Putting here \(\theta=\pi/2\) and \(n=2m\), we obtain
\[ \sum_{j=1}^{2m} \frac{1}{\cos^2 j\pi/(2m+1)} = 4m(m+1). \]
This identity is an analogue of the well-known identity (3) of M. Riesz \((^2)\).
Theorem 5. The interpolation process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)\equiv x\) at the nodes (6), converges uniformly on \([-1,1]\). Moreover,
\[ |H_{n+2}(f,x)-x|\leq 1/n. \]
The proof follows from the identity
\[ x-H_{n+2}(f,x)=\frac{1}{2(n+1)^2}\bigl(\cos\theta \sin^2(n+1)\theta-(n+1)\sin 2(n+1)\theta \sin\theta\bigr). \]
This theorem should be compared with the result of G. Szegő ([4], p. 349), according to which the Hermite–Fejér process constructed for \(f(x)\equiv x\) at the nodes
\[
\left\{\cos\frac{k\pi}{n+1}\right\}_{k=1}^{n},\qquad n=1,2,\ldots,
\]
diverges at the point \(x=1\).
Theorem 6. The process \(\{H_n(f)\}_{n=1}^{\infty}\), constructed for \(f(x)=|x|\) at the nodes (6), converges at the point \(x=0\). Moreover,
\[ |H_{n+2}(f,0)|\le C\ln n/n. \]
This theorem should be compared with the result in ([1]), according to which the Hermite–Fejér process constructed for \(f(x)=|x|\) at the nodes (2) diverges at the point \(x=0\).
Leningrad Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich
Received
6 IX 1968
REFERENCES
- D. L. Berman, DAN, 163, No. 3 (1965).
- M. Riesz, C. R., 158, 1152 (1914).
- D. L. Berman, DAN, 176, No. 2 (1967).
- G. Szegő, Orthogonal Polynomials, Moscow, 1962.