V. M. SOLODOV

ON THE ERROR OF NUMERICAL INTEGRATION

(Presented by Academician A. A. Dorodnitsyn on 9 VII 1962)

In this paper an unimprovable, in the sense of order, estimate is obtained for the error of numerical integration for a certain class of functions. The methods for obtaining the results are the same as in the works of N. M. Korobov \((^{1,2})\).

The functions \(f(x_1,\ldots,x_s)\) are assumed to be periodic, with period equal to one in each of the variables, and to be expandable in an absolutely convergent Fourier series:

\[ f(x_1,\ldots,x_s)= \sum_{m_1,\ldots,m_s=-\infty}^{\infty} c(m_1,\ldots,m_s)\exp\,[2\pi i(m_1x_1+\cdots+m_sx_s)]. \tag{1} \]

If the Fourier coefficients satisfy the inequality

\[ |c(m_1,\ldots,m_s)|\leq(\overline m_1\cdots \overline m_s)^{-\alpha} \qquad (\alpha>1,\ \overline m=\max(1,|m|)) \]

or the inequality

\[ |c(m_1,\ldots,m_s)|\leq \overline m_1^{-\alpha}(\overline m_2\cdots \overline m_s)^{-\alpha_1}, \qquad (\alpha,\alpha_1>1), \]

then we shall say that the function belongs, respectively, to the class \(E_s^\alpha\) or to the class \(E_s^{\alpha}\).

Lemma 1. The number of solutions of the inequality

\[ \overline m_1\cdots \overline m_s [\ln(1+\overline m_2)\cdots \ln(1+\overline m_s)]^2 \leq q \tag{2} \]

in integers \(m_1,\ldots,m_s\) does not exceed \(c_1(s)q\).

Proof by induction.

Lemma 2. Let \(p>s\) be a prime. There exists an integer \(a\) \((1<a<p)\) such that the congruence
\[ m_1+am_2+\cdots+a^{s-1}m_s\equiv 0 \pmod p \]
has no solutions, except the trivial one, in the domain (2) with \(q\leq [s c_1(s)]^{-1}p\).

Proof. Introduce the notation

\[ \delta_p(n)= \begin{cases} 1, & \text{if } n\equiv 0 \pmod p,\\ 0, & \text{if } n\not\equiv 0 \pmod p. \end{cases} \tag{3} \]

The lemma asserts that for some \(a\)

\[ \sum_{(2)}' \delta_p(m_1+am_2+\cdots+a^{s-1}m_s)=0, \]

where \(\sum_{(2)}'\) denotes summation over the domain (2), excluding the set \((m_1,\ldots,m_s)=(0,\ldots,0)\). We choose \(a\) from the condition

\[ \sum_{(2)}' \delta_p(m_1+am_2+\cdots+a^{s-1}m_s) = \min_{1\leq z<p} \sum_{(2)}' \delta_p(m_1+zm_2+\cdots+z^{s-1}m_s). \tag{4} \]

It is easy to show that the right-hand side of (4) is equal to 0. Indeed,

\[ \min_{1\le z\le p}\sum_{(2)}' \delta_p(m_1+zm_2+\cdots+z^{s-1}m_s) \le \frac1p\sum_{z=1}^{p}\sum_{(2)}' \delta_p(m_1+\cdots+z^{s-1}m_s) \le \]

\[ \le \frac{s-1}{p}\sum_{(2)}'1<1. \]

Corollary. Obviously, the assertion of Lemma 2 remains valid if the domain (2) is replaced by the domain

\[ \overline m_1(\overline m_2\ldots \overline m_s)^{1+\varepsilon}\le q\le c_2(\varepsilon,s)p,\qquad (\varepsilon>0). \]

Lemma 3. For \(\alpha>1,\ \varepsilon>0\), the estimate

\[ \sum_{\overline m_1(\overline m_2\ldots \overline m_s)^{1+\varepsilon}>q} \overline m_1^{-\alpha}(\overline m_2\ldots \overline m_s)^{-\alpha(1+\varepsilon)} \le c_3(\alpha,s,\varepsilon)q^{-(\alpha-1)}. \tag{5} \]

is valid.

Proof by induction.

Theorem 1. Let \(a\) be chosen as in Lemma 2. For functions \(f\in \overline E_s^\alpha\) the quadrature formula

\[ \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s - \frac1p\sum_{k=1}^{p} f\left(\frac1p k,\frac ap k,\ldots,\frac{a^{s-1}}p k\right) =R, \]

where \(|R|\le c(\alpha,\alpha_1,s)p^{-\alpha}\), is valid.

Proof. According to (1) and (3), the error of numerical integration can be written in the following form:

\[ |R|= \left| \frac1p \sum_{m_1,\ldots,m_s=-\infty}^{\infty}{}' c(m_1,\ldots,m_s) \sum_{k=1}^{p} \exp\left[2\pi i\,\frac{m_1+\cdots+a^{s-1}m_s}{p}\,k\right] \right| \le \]

\[ \le \sum_{m_1,\ldots,m_s=-\infty}^{\infty}{}' \overline m_1^{-\alpha} (\overline m_2\ldots \overline m_s)^{-\alpha\alpha_1} \delta_p(m_1+am_2+\cdots+a^{s-1}m_s), \tag{6} \]

where \(\sum'\) denotes summation over the sets \((m_1,\ldots,m_s)\ne(0,\ldots,0)\).

Applying Abel’s transformation successively with respect to each of the summation variables, we obtain

\[ |R|<(\alpha\alpha_1)^s \sum_{m_1,\ldots,m_s=1}^{\infty} m_1^{-\alpha-1}(m_2\ldots m_s)^{-\alpha\alpha_1-1} \times \]

\[ \times \sum_{|k_1|\le m_1,\ldots,|k_s|\le m_s}{}' \delta_p(k_1+ak_2+\cdots+a^{s-1}k_s). \tag{7} \]

Let us estimate from above the quantity

\[ \sum_{|k_1|\le m_1,\ldots,|k_s|\le m_s}{}' \delta_p(k_1+ak_2+\cdots+a^{s-1}k_s). \]

Obviously, it can be represented as a sum of \(2^s\) terms, in each of which every summation variable has constant sign. All terms are estimated in the same way. Therefore we consider only one of them:

\[ \sigma= \sum_{k_1=1}^{m_1}\cdots\sum_{k_s=1}^{m_s} \delta_p(k_1+ak_2+\cdots+a^{s-1}k_s). \]

By virtue of the consequence of Lemma 2, \(\sigma>0\) only when

\[ m_1(m_2\ldots m_s)^{\alpha_1} > m_1(m_2\ldots m_s)^{\alpha_1+\frac1\alpha-\frac{\alpha_1}{\alpha}} >q\ge c_2(\alpha,\alpha_1,s)\,p . \]

Define \(r,\rho\), and \(n\) from the conditions:

\[ m_1(m_2\ldots m_{r-1})^{\alpha_1} <q\le m_1(m_2\ldots m_r)^{\alpha_1}, \]

\[ q=m_1(m_2\ldots m_{r-1}\rho)^{\alpha_1}, \]

\[ [\rho]\,n<m_r\le[\rho]\,(n+1), \]

where \(n\) is an integer and \([\rho]\) is the integer part of \(\rho\). Then

\[ \sigma\le \sum_{k_s=1}^{m_s}\cdots \sum_{k_{r+1}=1}^{m_{r+1}} \sum_{k'_0=0}^{n} \left[ \sum_{k_r=k[\rho]+1}^{(k+1)[\rho]} \sum_{k_{r-1}=1}^{m_{r-1}}\cdots \sum_{k_1=1}^{m_1} \delta_p(k_1+ak_2+\ldots+a^{s-1}k_s) \right]. \]

The sum in square brackets does not exceed 1. Indeed, suppose it were greater than 1. This would mean that there exist at least two distinct solutions of the congruence
\(k_1+ak_2+\ldots+a^{s-1}k_s\equiv0\pmod p\), i.e.

\[ \delta_p(k'_1+\ldots+a^r k'_r+a^{r+1}k_{r+1}+\ldots+a^{s-1}k_s)=1 \]

and

\[ \delta_p(k''_1+\ldots+a^r k''_r+a^{r+1}k_{r+1}+\ldots+a^{s-1}k_s)=1. \]

In that case we would also have

\[ \delta_p(k'_1-k''_1+(k'_2-k''_2)a+\ldots+(k'_r-k''_r)a^r)=1. \]

But then

\[ \overline{(k'_1-k''_1)\,[(k'_2-k''_2)\ldots(k'_r-k''_r)]}^{\,\alpha_1} \le m_1(m_2\ldots m_{r-1}\rho)^{\alpha_1}=q, \]

which contradicts the choice of \(a\). Consequently,

\[ \sigma\le \sum_{k_s=1}^{m_s} \sum_{k_{r+1}=1}^{m_{r+1}} \sum_{k=0}^{n}1 = m_s\ldots m_{r+1}(n+1) \frac{(\rho m_{r-1}\ldots m_2)^{\alpha_1}m_1}{q} \le \]

\[ \le 4^{\alpha_1}\frac{m_1(m_2\ldots m_s)^{\alpha_1}}{q}. \]

Hence, by virtue of (7) and (5), we obtain

\[ |R|\le c_3(\alpha,\alpha_1,s) \sum_{m_1(m_2\ldots m_s)^{\alpha_1+\frac1\alpha-\frac{\alpha_1}{\alpha}}>q} \frac{m_1(m_2\ldots m_s)^{\alpha_1}} {q\,m_1^{\alpha+1}(m_2\ldots m_s)^{\alpha\alpha_1+1}} \le c(\alpha,\alpha_1,s)p^{-\alpha}. \]

The theorem is proved.

In Theorem 1 it was assumed that the number of nodes of the quadrature formula coincides with the quantity \(p\), which determined the form of the grid. Thus, when the number of nodes was changed, all nodes changed. In contrast, in the following theorem we consider such quadrature formulas in which increasing the number of nodes by one leads only to the addition of one new node to the old ones. In lectures by N. N. Chentsov at Moscow University it was proved that, for the error of quadrature formulas of this type, one cannot obtain an estimate better than \(R=O(N^{-1})\), where \(N\) is the number of nodes.

Theorem 2. Let \(a\) be chosen as in Lemma 2; let \(N>0\) be an arbitrary integer not exceeding \(p\). For functions \(f\in E^\alpha_{s-1}\) the quadrature formula

\[ \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_{s-1})\,dx_1\ldots dx_{s-1} - \frac1N\sum_{k=1}^{N} f\!\left(\frac{a}{p}k,\ldots,\frac{a^{s-1}}{p}k\right) =R, \tag{8} \]

where

\[ |R|\le \frac{c_4(\alpha,s)}{N}. \]

Proof. According to (6),

\[ |R|\ll \frac{1}{N} \sum_{m_1,\ldots,m_{s-1}=-\infty}^{\infty}{}' (\overline m_1\ldots \overline m_{s-1})^{-\alpha} \left|\sum_{k=1}^{N} \exp\left[\frac{am_1+\ldots+a^{s-1}m_{s-1}}{p}\,k\right]\right|. \]

Using the well-known estimate of a trigonometric sum, we obtain

\[ |R|\ll \frac{2}{N} \sum_{m_1,\ldots,m_{s-1}=-\infty}^{\infty}{}' (\overline m_1,\ldots,\overline m_{s-1})^{-\alpha} \min\left(N,\frac{1}{\left\langle\!\left\langle \frac{am_1+\ldots+a^{s-1}m_{s-1}}{p} \right\rangle\!\right\rangle}\right), \tag{9} \]

where \(\langle\!\langle \alpha\rangle\!\rangle\) is the distance from \(\alpha\) to the nearest integer. There always exists such an \(m_0\) that
\(am_1+\ldots+a^{s-1}m_{s-1}\equiv m_0\pmod p\) and
\(|m_0|\le \frac12(p-1)=p_1\). Obviously, the following equality holds:

\[ \left\langle\!\left\langle \frac{am_1+\ldots+a^{s-1}m_{s-1}}{p} \right\rangle\!\right\rangle^{-1} = \sum_{m_0=-p_1}^{p_1} \left\langle\!\left\langle \frac{m_0}{p}\right\rangle\!\right\rangle^{-1} \delta_p(am_1+\ldots+a^{s-1}m_{s-1}-m_0). \]

Since

\[ \left\langle\!\left\langle \frac{m_0}{p}\right\rangle\!\right\rangle = \frac{|m_0|}{p}, \]

it follows from (9) that

\[ \begin{aligned} |R| &\ll \frac{2p}{N} \sum_{m_1,\ldots,m_{s-1}=-\infty}^{\infty}{}' \sum_{m_0=-p_1}^{p_1} \frac{\delta_p(am_1+\ldots+a^{s-1}m_{s-1}-m_0)} {m_0(\overline m_1\ldots \overline m_{s-1})^\alpha} \\ &\ll \frac{p^{1+\varepsilon}}{N} \sum_{m_1,\ldots,m_{s-1}=-\infty}^{\infty}{}' \sum_{m_0=-p_1}^{p_1} \frac{\delta_p(am_1+\ldots+a^{s-1}m_{s-1}-m_0)} {m_0^{1+\varepsilon}(\overline m_1\ldots \overline m_{s-1})^\alpha}, \end{aligned} \tag{10} \]

where \(0<\varepsilon<\alpha-1\). Applying the result of Theorem 1 to (10), we obtain assertion (8).

Remark. The nodes obtained in Theorem 2 can be used to construct quadrature formulas with different weights for \(N\le p\). Namely, by the method of paper \({}^{3}\), for functions of the class \(E_s^\alpha\), where \(\alpha\) is not an integer, one can construct such formulas whose error has order \(N^{-[\alpha]}\).

Computing Center
Academy of Sciences of the USSR

Received
5 VII 1962

REFERENCES

\({}^{1}\) N. M. Korobov, DAN, 124, No. 6, 1207 (1959).
\({}^{2}\) N. M. Korobov, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 60, 195 (1961).
\({}^{3}\) C. B. Haselgrove, Math. Comput., 15, No. 76, 323 (1961).