Mathematics

Ya. L. Geronimus

On the Order of the Degree of Exactness of Chebyshev’s Quadrature Formula

(Presented by Academician S. L. Sobolev on 26 V 1969)

I. Denote by \(M_n\) the degree of exactness of Chebyshev’s quadrature formula with \(n\) nodes

\[ \int_{-1}^{1} p(x) f(x)\,dx = \frac{1}{n}\sum_{k=1}^{n} f(x_k), \qquad -1<x_1<x_2<\cdots<x_n<1, \tag{1} \]

i.e., the highest degree of a polynomial for which it is valid; the weight function \(p(x)\ge 0\) is assumed summable on the interval \([-1,+1]\).

In the present note we set ourselves the goal of finding estimates for the order, relative to \(n\), of the quantity \(M_n\), assuming that it increases without bound together with \(n\); here we use methods of S. N. Bernstein \((^1)\) and N. I. Akhiezer \((^2)\).

II. Introduce the notation

\[ t(\theta)=\pi p(-\cos\theta)|\sin\theta|, \qquad -\pi\le \theta\le \pi, \qquad x=-\cos\theta, \tag{2} \]

and impose on the behavior of the function \(t(\theta)\) the following restrictions of a local character:

A. The function \(t(\theta)\) is continuous at the point \(\theta=0\).

B. On the interval \([0,\varepsilon]\), where \(\varepsilon>0\) is a fixed small quantity, the function \(t(\theta)\) is positive almost everywhere.

C. The function \(t(\theta)\) does not decrease on the interval \([0,\varepsilon]\).

By \(2m-1\) denote the greatest odd number not exceeding \(M_n\), and we shall assume \(m\) so large that

\[ 0<\theta_1^{(m)}<\theta_2^{(m)}<\varepsilon; \]

here \(\{x_k^{(m)}=-\cos\theta_k^{(m)}\}_1^m\) are the roots of the polynomial of degree \(m\), orthogonal on the interval \([-1,+1]\) with respect to the weight function \(p(x)\).

The inequality

\[ \frac{2m}{n}\le \frac{2m\theta_2^{(m)}}{\pi}\,t(\theta_2^{(m)}); \tag{3} \]

holds; if, moreover, the function \(t(\theta)\) is continuous on \([0,\varepsilon]\), then *

\[ \frac{2m}{n} \le C_1\omega_2\!\left(\frac{1}{m};t\right) +\gamma_m\left|t(\theta_1^{(m)})-t(0)\right| +\gamma_m t(0), \tag{4} \]

where \(\omega_2(\delta;t)\) is the modulus of smoothness of the function \(t(\theta)\) on the interval \([0,\varepsilon]\), and

\[ \frac{8}{3}\le \gamma_m=\frac{4(2m^2+1)}{3m^2}\le 4. \]

Hence there follows a very simple result: if \(t(0)<3/8\), then

\[ \overline{\lim}_{n\to\infty} M_n/n<1; \]

thus, the condition \(t(0)\ge 3/8\) is necessary (but not sufficient) in order that, for unboundedly increasing values of \(n\), one have \(M_n=n\).

\[ \text{* } C,\ C_1,\ C_2,\ldots \text{ are constants independent of } m \text{ and } n. \]

III. In all that follows we shall assume that \(t(0)=0\); in order to estimate the order of the quantity \(M_n\), we need to find an estimate for the quantity \(\theta_2^{(m)}\) for unboundedly increasing values of \(m\).

Theorem. There is an estimate \(\theta_2^{(m)} \leq C\delta_m\), where the quantity \(\delta_m\) can be found as the root of the equation*

\[ \frac{1}{\delta}\lg \frac{2c_0}{a(\delta;t)}=m,\qquad c_0=\frac{1}{\pi}\int_0^\pi t(\theta)\,d\theta; \tag{5} \]

by \(a(\delta;t)\) is denoted the modulus of growth (see (4)) of the function \(t(\theta)\) on the interval \([-\varepsilon,+\varepsilon]\)

\[ a(\delta;t)=\inf \int_\varphi^{\varphi+\delta} t(\theta)\,d\theta,\qquad \varphi,\ \varphi+\delta\in[-\varepsilon,+\varepsilon], \tag{6} \]

which under our conditions is as follows:

\[ a(\delta;t)=\int_{-\delta/2}^{\delta/2} t(\theta)\,d\theta =2\int_0^{\delta/2} t(\theta)\,d\theta . \tag{7} \]

The inequalities (3), (4) can now be written as

\[ \frac{2m}{n}\leq \begin{cases} C_2 m\delta_m t(C\delta_m),\\ C_1\omega_2\left(\dfrac{1}{m};t\right)+\gamma_m t(C\delta_m). \end{cases} \tag{3′} \tag{4′} \]

Knowing the quantity \(\delta_m\), we can find an upper estimate for \(m\) as a function of \(n\), and consequently also an estimate for \(M_n<2m\).

IV. Let first the weight function \(p(x)\) have at the point \(x=-1\) a singularity of algebraic character

\[ p(x)=(1+x)^\gamma p_1(x),\qquad -\frac12<\gamma,\qquad 0<C_3\leq p_1(x)\leq C_4, \]
\[ x\in[-1,-1+\eta],\qquad \eta>0 \tag{8} \]

(where \(m\) is taken so large that \(-1<x_1^{(m)}<x_2^{(m)}<1+\eta\)). In this case, from (7) we find the estimate \(\delta_m\leq C_5\lg m/m\), after which formulas (3), (4) give

\[ \frac{2m}{n}\leq \begin{cases} C_6\lg m\left(\dfrac{\lg m}{m}\right)^{2\gamma+1},\\[6pt] C_1\omega_2\left(\dfrac{1}{m};t\right)+C_7\left(\dfrac{\lg m}{m}\right)^{2\gamma+1}. \end{cases} \tag{9} \]

The final determination of the estimate for \(M_n\) depends on the relative order of the quantities

\[ \lg m(\lg m/m)^{2\gamma+1},\qquad \omega_2(1/m;t),\qquad (\lg m/m)^{2\gamma+1}, \]

with respect to one another; the results are collected in Table 1**.

V. If additional restrictions are imposed on the behavior of the function \(p(x)\) in (8) on the entire interval \([-1,+1]\), then one can obtain the more precise estimate \(\delta_m\leq C_8\cdot 1/m\) (see (5, 6)); in particular, for this it suffices that the function \(p_1(x)\) in (8) be continuous and decreasing on the interval \([-1,+1]\), with \(-\frac12\leq\gamma\leq\frac12\) (condition C may then be discarded); using this estimate, we obtain the inequality

\[ 2m/n\leq C_1\omega_2(1/m;t)+C_5(1/m)^{2\gamma+1}; \tag{10} \]

if \((1/m)^{2\gamma+1}=o[\omega_2(1/m;t)]\), then we arrive at the estimate \(M_n\leq \varphi^{-1}(Cn)\); if, however, \(\omega_2(1/m;t)=o(1/m)^{2\gamma+1}\), then \(M_n\leq C_{10}n^{1/(2\gamma+2)}\),

* For the proof, see (2), Lemma 1.
** The function \(\varphi^{-1}\) is inverse to the function \(\varphi(m)=m/\omega_2(1/m;t)\).

VI. Let now the weight function \(p(x)\) have at the point \(x=-1\) a zero of logarithmic character

\[ p(x)=|\lg(1+x)|^{-\gamma}p_1(x),\qquad \gamma>0,\qquad x\in[-1,-1+\eta]; \tag{11} \]

it is not difficult to show that in this case we have the very same estimate \(\delta_m\le C_{11}\lg m/m\); the results are collected in Table 2.

Table 1

Table 2

In the case of a zero of exponential character

\[ p(x)=\exp\{-1/(1+x)^\gamma\}p_1(x),\qquad \gamma>0,\qquad x\in[-1,-1+\eta] \tag{12} \]

we find from (7) the estimate \(\delta_m\le C_{12}m^{-1/(2\gamma+1)}\); the final estimate for the degree of precision \(M_n\) is again as follows: \(M_n\le \varphi^{-1}(Cn)\).

VII. In conclusion we note the following: we have found an estimate for \(M_n\) by considering the behavior of the weight function \(p(x)\) near the point \(x=-1\); it is necessary to find an analogous estimate \(M_n'\) from consideration of its behavior near the point \(x=+1\), and then take the smaller of the quantities \(M_n\) and \(M_n'\).

Received
17 IX 1968

REFERENCES

1. S. N. Bernstein, DAN, 14, No. 6 (1937).
2. N. I. Akhiezer, Zhurn. Inst. Mat., AN USSR, No. 3 (1937).
3. Ya. L. Geronimus, Izv. AN SSSR, ser. matem., 27, No. 3, 529 (1963).
4. J. Shohat, Ann. Math., 18, 201 (1939).
5. P. Erdös, P. Turan, Ann. Math., 39, No. 4 (1938).
6. G. Freud, Acta Math. Acad. Sci. Hung., 4, No. 3–4 (1953).