MATHEMATICS

D. L. BERMAN

CONVERGENCE OF THE LAGRANGE INTERPOLATION PROCESS CONSTRUCTED FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND FUNCTIONS OF BOUNDED VARIATION

(Presented by Academician V. I. Smirnov on 25 VII 1956)

Let a triangular matrix of nodes be given

[
\begin{gathered}
x_1^{(1)}
\
x_1^{(2)}\quad x_2^{(2)}
\
\cdots
\
x_1^{(n)}\quad x_2^{(n)}\quad \ldots\quad x_n^{(n)}
\
\cdots
\
-1 \leq x_1^{(n)} < x_2^{(n)} < \cdots < x_n^{(n)} \leq 1
\quad (n=1,2,\ldots).
\end{gathered}
\tag{1}
]

Introduce the following notation:

[
\Delta_n =
\max_{k=1,2,\ldots,n-1}
\left[\left(x_{k+1}^{(n)}-x_k^{(n)}\right)\right].
]

Denote by (\mathfrak{S}(a,b)) the number of nodes in the (n)-th row of the matrix (1) satisfying the inequalities (a \leq x_j^{(n)} \leq b) (or (b \leq x_j^{(n)} \leq a)). Let

[
\omega_n(x)=\prod_{j=1}^{n}\left(x-x_j^{(n)}\right)\quad (n=1,2,\ldots);
]

[
l_j^{(n)}(x)=
\frac{\omega_n(x)}
{\left(x-x_j^{(n)}\right)\omega_n'\left(x_j^{(n)}\right)}
\quad (j=1,2,\ldots,n;\ n=1,2,\ldots);
]

[
\lambda_j^{(n)}(x)=\sum_{k=j}^{n} l_k^{(n)}(x).
]

For any function (f(x)) defined on the segment ([-1,1]), one can construct the Lagrange interpolation polynomial (L_n(f,x))

[
L_n(f,x)=\sum_{k=1}^{n} f\left(x_k^{(n)}\right)l_k^{(n)}(x).
]

Recently V. I. Krylov (¹) proved the theorem:

In order that, for every absolutely continuous function (f(x)), the relation

[
L_n(f,x)\to f(x),\qquad n\to\infty,
\tag{2}
]

hold uniformly on the segment ([-1,1]), it is necessary and sufficient that there exist a number (A) such that, for all (n), (j), and (x\in[-1,1]), the inequality

[
\left|\lambda_j^{(n)}(x)\right|\leq A
\tag{3}
]

be satisfied.

Han ((^2)) and S. M. Lozinskii ((^3)) proved the theorem:

For the relation (2) to hold at the point (x) for every function (f(x)) of bounded variation on ([-1,1]) and continuous at the point (x \in [-1,1]), it is necessary and sufficient that the following conditions hold:

1) At the point (x) the inequality
[
\left|\lambda_j^{(n)}(x)\right| \leq A(x), \qquad j=1,2,\ldots,n,\quad n=1,2,\ldots,
\tag{4}
]
holds, where the finite number (A(x)) depends only on (x).

2) If (t \in [-1,1]) and is distinct from the nodes ({x_k^{(n)}}{k=1}^n), (n=1,2,\ldots), then the equalities
[
\limt} l_k^{(n)}(x)=0 \qquad \text{for } t>x.}\sum_{x_k^{(n)
\tag{5}
]

Using the indicated theorems, V. I. Krylov ((^4)) established the convergence of the Lagrange interpolation process constructed at the nodes of P. L. Chebyshev
[
x_k^{(n)}=\cos \frac{2n-2k+1}{2n},\qquad k=1,2,\ldots;\quad n=1,2,\ldots,
]
in the case when (f(x)) is absolutely continuous or has bounded variation and is continuous at the given point of ([-1,1]).

In the present note the theorems of V. I. Krylov ((^4)) are extended to a certain broad class of node matrices (1), of which the nodes of P. L. Chebyshev are a particular case.

Theorem 1. Let the matrix (1) satisfy the conditions:

A. At each point (x \in [-1,1]) the inequalities hold:
[
\text{if } x_k^{(n)}<x_{k+1}^{(n)}\leq x
\qquad
\left|l_k^{(n)}(x)\right|\leq \left|l_{k+1}^{(n)}(x)\right|
\qquad (n=n_0,n_0+1,\ldots);
]
[
\text{if } x\leq x_k^{(n)}<x_{k+1}^{(n)}
\qquad
\left|l_k^{(n)}(x)\right|\geq \left|l_{k+1}^{(n)}(x)\right|
\qquad (n=n_0,n_0+1,\ldots).
]

B. There exists a nonnegative decreasing function (\varphi(h)), satisfying the condition (\varphi(h)\to 0) as (h\to\infty), such that, if (\mathcal{E}(x,x_k^{(n)})=h), then
[
\left|l_k^{(n)}(x)\right|\leq N\varphi(h)
\qquad (n=n_0,n_0+1,\ldots),\quad -1\leq x\leq 1,
]
where (N) is a finite nonnegative number.

Then the Lagrange interpolation process (L_n(f,x)), constructed for an absolutely continuous function (f(x)), converges uniformly on ([-1,1]) to (f(x)).

Proof. Obviously, it suffices to prove the validity of inequality (3). Let (x_p^{(n)}<x<x_{p+1}^{(n)}), and let (j<p)*; then
[
\left|\lambda_j^{(n)}(x)\right|
\leq
\left|\sum_{k=j}^{p} l_k\right|
+
\left|\sum_{k=p+1}^{n} l_k\right|,
\qquad
l_k=l_k^{(n)}(x).
\tag{6}
]
Since (\sum_{k=j}^{p} l_k) is estimated in exactly the same way as (\sum_{k=p+1}^{n} l_k), it is enough to consider only (\sum_{k=p+1}^{n} l_k). Obviously,
[
\left|\sum_{k=p+1}^{n} l_k\right|
\leq
|l_{p+1}+l_{p+2}|+|l_{p+3}+l_{p+4}|+\cdots
\tag{7}
]

[
\text{* In the opposite case the arguments are simplified.}
]

It is easy to see that, for (k>p+1), (\operatorname{sign} l_k(x)=-\operatorname{sign} l_{k+1}(x)). Therefore, in accordance with property A of the matrix (1), inequality (7) can be written in the form

[
\left|\lambda_{p+1}^{(n)}\right|
\le (|l_{p+1}|-|l_{p+2}|)+(|l_{p+3}|-|l_{p+4}|)+\cdots \le
]

[
\le |l_{p+1}|-(|l_{p+2}|-|l_{p+3}|)-\cdots .
\tag{8}
]

If we again use property A of the matrix (1), then from (8) it follows that

[
\left|\lambda_{p+1}^{(n)}(x)\right|\le |l_{p+1}|.
\tag{9}
]

Taking into account property B of the matrix (1), by virtue of (9) one can obtain the inequality

[
\left|\lambda_{p+1}^{(n)}(x)\right|\le N\varphi(1).
\tag{10}
]

Thus, from (6) and (9) it follows that

[
\left|\lambda_j^{(n)}(x)\right|\le 2N\varphi(1)<\infty,\qquad
x\in[-1,1],\qquad
j=1,2,\ldots,n;\quad n=n_0,\ n_0+1,\ldots
]

Theorem 1 is proved.

Remark. Theorem 1 can be formulated in a local form. In this case it is sufficient that conditions A and B be satisfied only at the given point of ([-1,1]).

Theorem 2. Let the (n)-th row of the matrix (1) be composed of the roots of the Jacobi polynomials
[
Y_n^{(\alpha_n,\beta_n)}(x),
]
where
[
-1\le \alpha_n,\ \beta_n\le -\lambda<0,\qquad n=1,2,\ldots,*,
]
where (\lambda) is an arbitrarily small positive number.

Then, for any absolutely continuous function (f(x)), relation (2) holds uniformly on the segment ([-1,1]).

If the matrix (1) is composed of the roots of the Legendre polynomials ((\alpha_n=\beta_n=0)) and (f(x)) is absolutely continuous, then relation (2) holds at every point of the interval ((-1,1)). The convergence is uniform on any segment of the form ([-1+\varepsilon,\,1-\varepsilon]), (0<\varepsilon<1).

Proof. In my note ((^5)) it was proved that the matrix (1), composed of the roots of the polynomials
[
Y_n^{(\alpha_n,\beta_n)}(x),
]
where
[
-1\le \alpha_n,\ \beta_n\le -\lambda<0,\qquad n=1,2,\ldots,
]
satisfies conditions A and B of Theorem 1. Therefore the first part of Theorem 2 follows directly from Theorem 1. The second part of Theorem 2 follows from the local form of Theorem 1 and from the fact that the matrix (1), composed of the roots of the Legendre polynomials, satisfies conditions A and B of Theorem 1 in the local form ((^5)).

Theorem 3. Let the matrix (1) satisfy conditions A and B of Theorem 1.

Then, if the function (f(x)) has bounded variation on ([-1,1]), relation (2) holds at all points of continuity of (f(x)).

To prove Theorem 3 it is sufficient to prove the validity of the equalities (5). We shall prove the second of the equalities (5); the first equality is proved in the same way.

Let (x_s<t<x_{s+1}); then from inequality (9) it follows that

[
\left|\sum_{x_k^{(n)}>t} l_k(x)\right|
=
\left|\lambda_{s+1}^{(n)}(x)\right|
\le |l_{s+1}|.
]

[
{}^*\ \text{By definition }\quad
Y_n^{(-1,-1)}(x)=\int_{-1}^{x} P_{n-1}(t)\,dt,
\quad \text{where } P_n(t)\text{ is the Legendre polynomial of degree }n.
]

But, according to condition B,

[
|l_{s+1}| \leq N\varphi!\left(\frac{t-x}{\Delta_n}\right), \qquad t>x .
]

Therefore

[
\left| \sum_{x_k^{(n)}>t} l_k(x) \right|
\leq
N\varphi!\left(\frac{t-x}{\Delta_n}\right), \qquad t>x .
\tag{11}
]

Since (\Delta_n \to 0) ((^6)), it follows from inequality (11) and the property of the function (\varphi(h)) that (5) is fulfilled.

Theorem 4. Let the (n)-th row of the matrix (1) be composed of the zeros of the Jacobi polynomials (Y_n^{(\alpha_n,\beta_n)}(x)), (-1 \leq \alpha_n,\ \beta_n \leq 0,\ n=1,2,\ldots).

Then, if the function (f(x)) has bounded variation on ([-1,1]), relation (2) is fulfilled at all points of continuity of (f(x)) in ((-1,1)).

This theorem follows directly from Theorem 3 and from the results ((^5)).

It is clear that the results of V. I. Krylov ((^4)) are special cases of Theorems 2 and 4, when (\alpha_n=\beta_n=-1/2).

In conclusion I note that the method of proof presented here is of a more elementary character than the method of proof in ((^4)).

Novgorod State
Pedagogical Institute

Received
20 VII 1956

CITED LITERATURE

(^1) V. I. Krylov, DAN, 105, No. 2 (1955).
(^2) H. Hahn, Math. Zs., 1, 115 (1918).
(^3) S. M. Lozinskii, Uchen. zap. Leningr. univ., No. 55, 84 (1940).
(^4) V. I. Krylov, DAN, 107, No. 3 (1956).
(^5) D. L. Berman, DAN, 60, No. 3 (1948).
(^6) P. Erdös, P. Turán, Ann. of Math., 39, No. 4, 703 (1938).