MATHEMATICS

D. G. SANIKIDZE

ON THE DIVERGENCE OF INTERPOLATION PROCESSES

(Presented by Academician P. S. Aleksandrov, 30 I 1969)

Let an infinite triangular matrix of nodes be given

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

For every function \(f(x)\) defined on \([-1,+1]\), one can construct the Lagrange polynomial interpolating \(f(x)\) at the nodes of the \(n\)-th \((n=1,2,\ldots)\) row of matrix (1):

\[ L_{n-1}(f;x)=\sum_{k=1}^{n} l_{n,k}(x) f\bigl(x_k^{(n)}\bigr), \]

\[ l_{n,k}(x)= \frac{\omega_n(x)} {\bigl(x-x_k^{(n)}\bigr)\omega_n'\bigl(x_k^{(n)}\bigr)}, \qquad \omega_n(x)=\prod_{k=1}^{n}\bigl(x-x_k^{(n)}\bigr). \]

As is known, the sums

\[ \sum_{k=1}^{n} |l_{n,k}(x)| \quad (n=1,2,\ldots) \]

grow without bound for any system of nodes, and it is impossible to specify such a matrix (1) that, for every continuous function,

\[ L_{n-1}(f;x)\to f(x) \quad \text{as } n\to\infty . \tag{2} \]

Meanwhile, as was shown by V. I. Krylov \((^1)\), if the matrix (1) is Chebyshev, then (2) holds uniformly on \([-1,+1]\) for every function \(f\) absolutely continuous on \([-1,+1]\). D. L. Berman \((^2)\) extended V. I. Krylov’s results to a sufficiently broad class of matrices (1), of which the Chebyshev system of nodes is a special case.

In this connection, the following question is of some interest: will a positive result hold in the class \(ACG_*\) (\((^3)\), p. 333) of functions?

The answer to this question is negative.

Theorem. There exists no system of nodes guaranteeing the convergence of interpolation for all functions \(\in ACG_*\).

Proof. Following \((^4)\), for every \(f\in ACG_*\) the interpolation remainder can be represented in the form

\[ R_n(f;x)= \int_{-1}^{+1} \left[ E(x-t)-\sum_{k=1}^{n} l_{n,k}(x) E\bigl(x_k^{(n)}-t\bigr) \right] f'(t)\,dt, \]

\[ E(x)= \begin{cases} 1, & x \geq 0,\\ 0, & x < 0, \end{cases} \]

where the integral is understood in the Denjoy–Perron sense.

The further arguments are based on the following theorem of A. G. Djvaršeišvili \((^5)\).

Let \(\{g_n(x)\}\) be a sequence of functions locally monotone\(^*\) on \([-1,+1]\) such that

\[ \overline{\lim}_{n\to\infty}\left|\int_{-1}^{+1} g_n(x)\psi(x)\,dx\right|<L \qquad (n=1,2,\ldots) \tag{3} \]

for every summable function \(\psi(x)\) on \([-1,+1]\). Then the inequalities

\[ \left|\int_{-1}^{+1} g_n(x)\varphi(x)\,dx\right|\le M(\varphi) \qquad (n=1,2,\ldots) \]

can hold for every function \(\varphi(x)\) integrable in the Denjoy–Perron sense if and only if the total variations of the functions \(g_n(x)\) \((n=1,2,\ldots)\) are bounded in the aggregate:

\[ \operatorname*{Var}_{-1}^{+1} g_n(x)\le N \qquad (n=1,2,\ldots). \tag{4} \]

Since, for arbitrary \(x\) and \(n\), the expression

\[ F_n(t)=E(x-t)-\sum_{k=1}^{n} l_{n,k}(x)E\bigl(x_k^{(n)}-t\bigr) \]

is a piecewise constant function on \([-1,+1]\), it satisfies the condition of local monotonicity for every system of nodes (1). Moreover, as was noted above, there exist matrices of nodes such that

\[ \lim_{n\to\infty} R_n(f;x)=0 \]

uniformly on \([-1,+1]\) for every absolutely continuous function \(f\). Therefore, for such matrices of nodes, conditions (3) for the integrals

\[ \int_{-1}^{+1} F_n(t)f'(t)\,dt \qquad (n=1,2,\ldots) \]

are satisfied. On the other hand, for any \(x\),

\[ \operatorname*{Var}_{-1}^{+1}\sum_{k=1}^{n} l_{n,k}(x)E\bigl(x_k^{(n)}-t\bigr) = \sum_{k=1}^{n}|l_{n,k}(x)|, \]

whence it follows that conditions (4) for the sequence \(\{F_n(t)\}\) cannot be satisfied, whatever the matrix of nodes (1) may be. In view of this, according to the theorem of A. G. Dzhvarsheishvili, there exists a function \(f\in ACG_*\) such that the sequence \(\{R_n(f;x)\}\) will not be bounded. This proves the theorem.

REFERENCES

\[ {}^{1}\ \text{V. I. Krylov, DAN, 107, No. 3 (1956).} \qquad {}^{2}\ \text{D. L. Berman, DAN, 112, No. 1 (1957).} \]

\[ {}^{3}\ \text{S. Saks, Theory of the Integral, Moscow, 1949.} \qquad {}^{4}\ \text{V. I. Krylov, DAN, 105, No. 2 (1955).} \]

\[ {}^{5}\ \text{A. G. Dzhvarsheishvili, Communications of the Academy of Sciences of the Georgian SSR, 17, No. 4 (1956).} \]

\(^*\) For each point \(x\in[-1,+1]\) there exist intervals \((x-\delta,x)\) and \((x,x+\delta)\) on which \(g_n(x)\) is monotone.