MATHEMATICS

I. F. SHARYGIN

ON THE APPLICATION OF NUMBER-THEORETIC METHODS OF INTEGRATION IN THE CASE OF NONPERIODIC FUNCTIONS

(Presented by Academician S. L. Sobolev, 3 XII 1959)

1. In numerous works devoted to the approximate computation of multiple integrals that have appeared recently, quadrature formulas have been constructed, in particular, for functions \(f(x_1,\ldots,x_s)\) belonging to the class \(E_s(\alpha)\), defined by the conditions:

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

\[ c(m_1,\ldots,m_s)=O\left((\overline m_1\cdots \overline m_s)^{-\alpha}\right), \qquad \overline m_i=\max(1,|m_i|),\qquad \alpha>1. \]

Let us denote by \(H_s(\alpha,c)\) the class of functions defined by the conditions

\[ \left| \frac{\partial^n f(x_1,\ldots,x_s)} {\partial x_1^{\gamma_1}\cdots \partial x_s^{\gamma_s}} \right| <c,\quad 0\leq n\leq \alpha s,\qquad 0\leq \gamma_i\leq \alpha,\qquad \gamma_1+\cdots+\gamma_s=n. \]

Periodic functions from \(H_s(\alpha,c)\) also belong to \(E_s(\alpha)\). However, it is possible to construct quadrature formulas acting on the whole class \(H_s(\alpha,c)\).

Theorem 1. Let \(f(x_1,\ldots,x_s)\) belong to \(H_s(\alpha,c)\), \(\alpha>1\); let \(\tau_\alpha(z)\) be a certain function satisfying the conditions:

1) \[ 0=\tau_\alpha(0)<\tau_\alpha(z')<\tau_\alpha(z'')<\tau_\alpha(1)=1,\quad 0<z'<z''<1; \]

2) \[ \left|\tau_\alpha^{(k)}(z)\right|<A,\quad 0\leq k\leq \alpha+1; \]

3) \[ \tau_\alpha^{(k)}(0)=\tau_\alpha^{(k)}(1)=0,\quad 1\leq k\leq \alpha. \]

Then there exist integers \(a_1,\ldots,a_s\), \(a_i=a_i(N)\), such that

\[ R= \left| \frac{1}{N}\sum_{k=1}^{N} f\left[ \tau_\alpha\left(\left\{\frac{ka_1}{N}\right\}\right), \ldots, \tau_\alpha\left(\left\{\frac{ka_s}{N}\right\}\right) \right] \tau_\alpha'\left(\left\{\frac{ka_1}{N}\right\}\right) \cdots \tau_\alpha'\left(\left\{\frac{ka_s}{N}\right\}\right) - \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s \right| = O\left(\frac{\ln^{\alpha s}N}{N^\alpha}\right)^{*}. \]

Proof. In the integral

\[ \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s \]

we make the substitution \(x_i=\tau_\alpha(z_i)\). The integrand after this substitution obviously turns out to belong to the class \(E_s(\alpha)\), and therefore the result being proved follows from the corresponding theorems \((^1,^2)\).

\(*\) \(\{x\}\) is the fractional part of \(x\).

Remark. Let \(f(x_1,\ldots,x_s)\) have a singularity on the hyperplane \(x_1=x_1^0,\ldots,x_\nu=x_\nu^0;\ 1\leqslant \nu\leqslant s\);

\[ \frac{\partial^{\alpha\nu} f(x_1,\ldots,x_s)} {\partial x_1^\alpha\cdots \partial x_\nu^\alpha} = O\left(\frac{1}{r^{\beta+\alpha\nu}}\right), \quad 0<\beta<\nu, \]

\[ r=\sqrt{(x_1-x_1^0)^2+\cdots+(x_\nu-x_\nu^0)^2}. \]

Suppose that \(f\) has the mixed derivatives occurring in the definition of the class \(H_s(\alpha,c)\), continuous in the remaining part of the unit cube. Then one can compute integrals of such functions by introducing \(\tau_{\alpha,i}(z)\) satisfying the condition
\[ \tau_{\alpha,i}^{(k)}(z_i^0)=0,\quad 1\leqslant k\leqslant \alpha; \]
\[ \tau_{\alpha,i}^{(\alpha)}(z_i)=O\bigl((z_i-z_i^0)^{b-\alpha}\bigr), \quad b=\frac{\alpha\nu}{\nu-\beta}, \quad \tau_{\alpha,i}(z_i^0)=x_i^0,\quad 1\leqslant i\leqslant \nu \, *. \]

1. Let \(f(x_1,\ldots,x_s)\) be given on the unit cube and be different from zero on some parallelepiped whose faces are parallel to the coordinate planes. Suppose that \(f(x_1,\ldots,x_s)\) on this parallelepiped satisfies the conditions defining the function class \(H_s(1,c)\). Extend \(f(x_1,\ldots,x_s)\) to the whole space by setting

\[ f(x_1,\ldots,x_s)=f(\{x_1\},\ldots,\{x_s\}). \]

Introduce

\[ f_\Delta(x_1,\ldots,x_s) = \int_{-\Delta}^{\Delta}\cdots\int_{-\Delta}^{\Delta} f[(x_1+\xi_1),\ldots,(x_s+\xi_s)]\,d\xi_1\cdots d\xi_s, \]

\[ \int_0^1\cdots\int_0^1 f_\Delta(x_1,\ldots,x_s)\,dx_1\cdots dx_s = \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s. \tag{1} \]

If \(c(m_1,\ldots,m_s)\) and \(c_\Delta(m_1,\ldots,m_s)\) are the Fourier coefficients of the functions \(f(x_1,\ldots,x_s)\) and \(f_\Delta(x_1,\ldots,x_s)\), then, obviously,

\[ |c(m_1,\ldots,m_s)|\leqslant \frac{c'}{m_1\cdots m_s}, \tag{2} \]

\[ |c_\Delta(m_1,\ldots,m_s)| \leqslant \frac{c'}{m_1\cdots m_s} \left|\frac{\sin 2\pi m_1\Delta}{2\pi m_1\Delta}\right| \cdots \left|\frac{\sin 2\pi m_s\Delta}{2\pi m_s\Delta}\right|, \tag{3} \]

\[ c'=c'(c,s). \]

Consider

\[ R^N(z,f) = \left| \frac{1}{N}\sum_{k=1}^{N} f\left(\frac{k}{N},\frac{kz}{N},\ldots,\frac{kz^{s-1}}{N}\right) - \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s \right|; \tag{4} \]

\[ R_\Delta^N(z,f) = \left| \frac{1}{N}\sum_{k=1}^{N} f_\Delta\left(\frac{k}{N},\frac{kz}{N},\ldots,\frac{kz^{s-1}}{N}\right) - \right. \]

\[ \left. - \int_0^1\cdots\int_0^1 f_\Delta(x_1,\ldots,x_s)\,dx_1\cdots dx_s \right|; \tag{5} \]

\[ r_\Delta(z,f) = \left| \frac{1}{N}\sum_{k=1}^{N} f\left(\frac{k}{N},\frac{kz}{N},\ldots,\frac{kz^{s-1}}{N}\right) - \frac{1}{N}\sum_{k=1}^{N} f_\Delta\left(\frac{k}{N},\frac{kz}{N},\ldots,\frac{kz^{s-1}}{N}\right) \right|. \tag{6} \]

\[ \underline{\hspace{2cm}} \]

* The possibility of computing integrals of functions having singularities by means of \(\tau_\alpha(z)\) was indicated by participants in the seminar on the number-theoretic Monte Carlo method at the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR.

Theorem 2. If \(N=p>s\) is prime, then there exists an integer \(a=a(p)\), \(1\leq a\leq p-1\), such that
\[ R^N(a,f)=O\left(\frac{\ln^s N}{N}\right). \]

Proof. From (1), (4), (5), and (6) we obtain
\[ R^N(z,f)\leq R_\Delta^N(z,f)+r_\Delta(z,f). \tag{7} \]

Let us estimate
\[ \min_{1\leq z\leq p-1}\ \max_{f\in H_s(\alpha,c)} R_\Delta^p(z,f). \]

Using the known estimates (4) and (3), we obtain
\[ \min_{1\leq z\leq p-1}\ \max_{f\in H_s(\alpha,c)} R_\Delta^p(z,f) \leq \max_{f\in H_s(\alpha,c)} \frac{1}{p-1}\sum_{z=1}^{p-1} R_\Delta^p(z,f) \leq \]
\[ \leq \frac{c'}{p-1} \sum_{m_1,\ldots,m_s=-\infty}^{\infty}{}' \frac{1}{\overline{m}_1\cdots \overline{m}_s} \left|\frac{\sin 2\pi m_1\Delta}{2\pi m_1\Delta}\right| \cdots \left|\frac{\sin 2\pi m_s\Delta}{2\pi m_s\Delta}\right| A(m_1,\ldots,m_s), \]
where
\[ 0\leq A(m_1,\ldots,m_s) \begin{cases} = p-1 & \text{if } d\equiv 0 \pmod p,\\ \leq s-1 & \text{if } d\not\equiv 0 \pmod p; \end{cases} \]
\(d\) is the greatest common divisor of \(m_1,\ldots,m_s\). The prime on the summation sign means that the value \(m_1=\cdots=m_s=0\) is excluded. Take \(\Delta=\dfrac{1}{p}\). Then the terms corresponding to tuples \(m_1,\ldots,m_s\) for which \(d\equiv 0 \pmod p\) vanish if at least one \(m_i\ne 0\). Consequently, for all terms different from zero,
\(A(m_1,\ldots,m_s)\leq s-1\). Thus,
\[ \min_{1\leq z\leq p-1}\ \max_{f\in H_s(\alpha,c)} R_\Delta^p(z,f) \leq \frac{(s-1)c'}{p-1} \left( 1+2\sum_{m=1}^{\infty}\frac{1}{m} \left|\frac{\sin 2\pi m\Delta}{2\pi m\Delta}\right| \right)^s; \]
\[ \left|\frac{\sin 2\pi m\Delta}{2\pi m\Delta}\right| \leq \begin{cases} 1, & \text{if } m<\dfrac{1}{\Delta}=p,\\[6pt] \dfrac{1}{m\Delta}, & \text{if } m\geq\dfrac{1}{\Delta}=p. \end{cases} \]

Consequently,
\[ \min_{1\leq z\leq p-1}\ \max_{f\in H_s(\alpha,c)} R_\Delta^p(z,f) \leq \]
\[ \leq \frac{(s-1)c'}{p-1} \left( 1+2\sum_{m=1}^{p-1}\frac{1}{m} +2\sum_{m=p}^{\infty}\frac{1}{m^2\Delta} \right)^s = O\left(\frac{\ln^s p}{p}\right). \tag{8} \]

Let us now estimate \(r_\Delta(z,f)\). If the point
\[ \left(\frac{k}{p},\left\{\frac{kz}{p}\right\},\ldots,\left\{\frac{kz^{s-1}}{p}\right\}\right) \]
is at a distance greater than \(\Delta\) from the surface of the parallelepiped, then
\[ \left| f\left(\frac{k}{p},\frac{kz}{p},\ldots,\frac{kz^{s-1}}{p}\right) - f_\Delta\left(\frac{k}{p},\frac{kz}{p},\ldots,\frac{kz^{s-1}}{p}\right) \right| \leq cs\Delta. \]

The number of points
\[ \left(\frac{k}{p},\left\{\frac{kz}{p}\right\},\ldots,\left\{\frac{kz^{s-1}}{p}\right\}\right) \]
that are at a distance not exceeding \(\Delta\) from the surface of the parallelepiped, for each \(z\), does not exceed \(\Delta p\cdot 2^s\). From what has been said there follows the following estimate for \(r_\Delta(z,f)\):
\[ r_\Delta(z,f)\leq \Delta(cs+2^s). \tag{9} \]

From (7), (8), (9) we obtain

\[ \min_{1\le z\le p-1}\ \max_{f\in H_s(\alpha,c)} R^p(z,f) =O\left(\frac{\ln^s p}{p}\right), \]

which proves Theorem 2.

Remark. It is not difficult to show that

\[ R^p(a_1,\ldots,a_s) = \left| \frac1p\sum_{k=1}^p f\left(\frac{ka_1}{p},\ldots,\frac{ka_s}{p}\right) - \right. \]

\[ \left. -\int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s \right| = O\left(\frac{\ln^s p}{p}\right), \]

where \(a_1,\ldots,a_s\) are the optimal coefficients determined in [2].

1. Let the equation be given

\[ \varphi(x_1,\ldots,x_s) = \int_0^{x_1}\cdots\int_0^{x_l}\int_0^1\cdots\int_0^1 K(y_1,\ldots,y_s,\ x_1,\ldots,x_s)\, \varphi(y_1,\ldots,y_s) \times \]

\[ \times\, dx_1\cdots dy_s + f(x_1,\ldots,x_s),\qquad l\ge 1, \tag{10} \]

where \(K(y_1,\ldots,y_s,x_1,\ldots,x_s)\in H_{2s}(\alpha,c_1)\), \(f(x_1,\ldots,x_s)\in H_s(\alpha,c_2)\). It is obvious that \(\varphi(x_1,\ldots,x_s)\in H_s(\alpha,c_3)\), \(c_3=c_3(c_1,c_2,s)\).

If \(K_1(y_1,\ldots,y_s,x_1,\ldots,x_s)=K(y_1,\ldots,y_s,x_1,\ldots,x_s)\) for \(0\le y_i\le x_i,\ 1\le i\le l\), and \(K_1(y_1,\ldots,y_s,x_1,\ldots,x_s)=0\) at the remaining points of the unit cube, then (in the notation adopted) equation (10) can be rewritten in the form

\[ \varphi(P)=\int_{G_s} K_1(Q,P)\varphi(Q)\,dQ+f(P). \tag{11} \]

Take in the unit \(s\)-dimensional cube \(G_s\) the points

\[ M_i=M\left(\frac{i}{N},\left\{\frac{ia}{N}\right\},\ldots, \left\{\frac{ia^{s-1}}{N}\right\}\right), \]

where \(a\) is defined in Theorem 2.

Theorem 3. If the quantities \(\widetilde{\varphi}(M_j)\) satisfy the system of linear equations

\[ \widetilde{\varphi}(M_j) = \frac1N\sum_{i=1}^{N} K_1(M_i,M_j)\,\widetilde{\varphi}(M_i) + f(M_j) \qquad (j=1,2,\ldots,N), \]

then the equality

\[ \varphi(M_j)=\widetilde{\varphi}(M_j)+ O\left(\frac{\ln^s N}{N}\right) \]

holds.

The proof of this theorem follows obviously from Theorem 2 and from [3].

The questions of the approximate solution of Volterra-type integral equations were also studied by Yu. N. Shakhov [5].

Moscow State University
named after M. V. Lomonosov

Received
20 XI 1959

CITED LITERATURE

1. N. S. Bakhalov, Vestn. MGU, No. 4 (1959).
2. N. M. Korobov, DAN, 128, No. 6 (1959).
3. N. M. Korobov, DAN, 128, No. 2 (1959).
4. N. M. Korobov, Vestn. MGU, No. 4 (1959).
5. Yu. N. Shakhov, DAN, 128, No. 6 (1959).