MATHEMATICS

Academician S. L. SOBOLEV

ON FORMULAS OF MECHANICAL CUBATURES IN \(n\)-DIMENSIONAL SPACE

Formulas of mechanical cubatures

\[ (l,\varphi)\equiv \int\limits_{\Omega}\varphi\,dx-\sum_{k=1}^{N} C_k\varphi\bigl(x^{(k)}\bigr)\cong 0, \tag{1} \]

where \(x\) is a point of a bounded \(n\)-dimensional domain \(\Omega\); \(C_k\) are coefficients; \(x^{(k)}\) are the nodes of the formula, give different accuracy for different classes of functions. We shall assume that the error \((l,\varphi)\) is equal to zero for polynomials of a certain degree \(m_1\). The domain \(\Omega\) will be assumed to have a piecewise smooth boundary.

Of special interest for the theory is the particular case of cubature formulas when the function \(\varphi\) is periodic with periods \(H\beta\), where \(H\) is the fundamental matrix of periods

\[ H=(\mathbf h_1,\mathbf h_2,\ldots,\mathbf h_n), \tag{2} \]

each period \(\mathbf h_k\) is a column vector

\[ \mathbf h_k= \begin{pmatrix} h_{1k}\\ h_{2k}\\ \ldots\\ h_{nk} \end{pmatrix}; \tag{3} \]

\(\beta\) is an integer column vector

\[ \beta= \begin{pmatrix} \beta_1\\ \beta_2\\ \ldots\\ \beta_n \end{pmatrix}, \qquad -\infty<\beta_k<+\infty . \tag{4} \]

In this case the domain of integration \(\Omega_0\) is the fundamental parallelepiped such that the system of all \(\Omega_\beta\), obtained by translating \(\Omega_0\) by \(H\beta\), covers the whole space \(R_n\) without intersection. In this case we put \(m_1=0\), i.e. we shall assume that formula (1) is valid for \(\varphi=1\), and hence

\[ \sum_{k=1}^{N} C_k=|\Omega_0|. \tag{5} \]

The principal problem of the theory of mechanical cubatures is the determination of

\[ \min_{C_k,x^{(k)}}\,[\max |(l,\varphi)|]=d(X,N) \tag{6} \]

for a given class \(X\) and a given number of points \(N\). The values \(C_k\) and \(x^{(k)}\) for which this minimax is attained give the optimal formula of mechanical cubatures. For \(n=1\) this problem was considered by S. M. Nikol’skii and his students. A survey of the results obtained is given in \((^3)\); the most important bibliography of the question is also included there.

Recently, cubature formulas for \(n>1\) have been studied by many authors \((^{7-13})\).

It is convenient to take as \(X\) the unit ball in some Banach space \(B\), in which \((l,\varphi)\) is linear. Let us note that in the space \(C\) of continuous functions in \(\Omega\) or \(\Omega_0\) there are no functionals of the form \((l,\varphi)\) that are small on the whole unit ball, although all of them are linear:

\[ \sup_{\|\varphi\|_C=1}|(l,\varphi)|=\left[|\Omega_0|+\sum_{k=1}^{N}|C_k|\right]. \tag{7} \]

Therefore, in \(C\) the main problem has no meaning.

In the present note we shall study the first half of the main problem, i.e. the finding of

\[ \max_X |(l,\varphi)|=d(C_k,x^{(k)}) \tag{8} \]

in the case when \(X\) is the unit ball in the space of functions whose derivatives of order \(m\) are square-integrable. The second half of the problem—the finding of \(\min d(C_k,x^{(k)})\)—is a problem on the minimum of a function of \((n+1)N\) variables, and we shall not touch upon it. At the same time we shall consider the spaces \(\widetilde W_2^{(m)}\) and \(\widetilde L_2^{(m)}\) of functions periodic in \(R_n\), with periods \(H\beta\), and for \(\Omega\) we take the parallelepiped \(\Omega_0\).

The norms in \(W_2^{(m)}\) and \(\widetilde W_2^{(m)}\) are given by the formulas \((^2)\)

\[ \|\varphi\|_{W_2^{(m)}}^2=\|\Pi\varphi\|_{S_{m-1}}^2+\|\varphi\|_{L_2^{(m)}}^2 =\|\Pi\varphi\|_{S_{m-1}}^2+D(\varphi), \tag{9} \]

\[ \|\varphi\|_{\widetilde W_2^{(m)}}^2= \left(\int_{\Omega_0}\varphi\,dx\right)^2+D(\varphi), \tag{10} \]

where \(\Pi\) is the projection operator from \(W_2^{(m)}\) into the space \(S_{m-1}\) of polynomials of degree \(m\), and \(L_2^{(m)}\) is the factor space \(W_2^{(m)}/S_{m-1}\). Here

\[ \|\varphi\|_{L_2^{(m)}}^2=D(\varphi)= \int_{\Omega}\sum_{|\alpha|=m}(D^\alpha\varphi)^2\,dx. \tag{11} \]

For the nonperiodic case we shall henceforth always assume \(m_1=m-1\), i.e.

\[ (l,\varphi)=0\quad \text{for } \varphi\in S_{m-1}. \tag{12} \]

If the operator \(\Pi\varphi\) is interpolatory, then the general case reduces to this one.

The following inequalities hold:

\[ |(l,\varphi)|\le K\|\varphi\|_{L_2^{(m)}}\le K\|\varphi\|_{W_2^{(m)}},\qquad |(l,\varphi)|\le K\|\varphi\|_{\widetilde L_2^{(m)}}\le K\|\varphi\|_{\widetilde W_2^{(m)}}. \tag{13} \]

Consider three problems:

Problem I. Find
\[ \max_{\|\varphi\|_{W_2^{(m)}}=1}(l,\varphi). \]

Problem II. Find
\[ \min_{(l,\varphi)=1}\|\varphi\|_{W_2^{(m)}}^2. \]

Problem III. Find
\[ \min H_\lambda(\varphi)=D(\varphi)+2\lambda(l,\varphi). \]

All three problems reduce to any one of them.

Let us consider Problem III by means of the direct method. In view of (9), \(H_\lambda(\varphi)\) has a finite exact lower bound:

\[ H_\lambda(\varphi)\ge [\sqrt{D(\varphi)}-\lambda K]^2-\lambda^2K^2\ge -\lambda^2K^2. \tag{14} \]

The identity

\[ \frac12 H_\lambda(u_k)+\frac12 H_\lambda(u_m)-H_\lambda\left(\frac{u_k+u_m}{2}\right) = D\left(\frac{u_k-u_m}{2}\right) \tag{15} \]

allows one to conclude that if \(u_k\) is a minimal sequence, then \(\Pi u_k\) will again be minimal and, moreover, fundamental, with the unique limit giving the solution of problem III. Further, from the identity

\[ H_\lambda(\varphi)=\frac{\lambda^2}{\lambda_1^2}H_{\lambda_1}\left(\frac{\lambda_1\varphi}{\lambda}\right) \tag{16} \]

one may conclude that the solutions of problem III for different \(\lambda\) differ from one another by a factor and are expressed by the formula

\[ u_\lambda=\lambda u_1. \tag{17} \]

Consideration of the function \(\psi(\mu)=H(\mu u_\lambda)\) leads to the conclusion that if

\[ H_\lambda(u_\lambda)=\min H_\lambda(u)=-d_\lambda(C_k,x^{(k)}), \tag{18} \]

then

\[ D(u_\lambda)=d_\lambda(C_k,x^{(k)});\qquad (l,u_\lambda)=-d_\lambda(C_k u^{(k)}). \tag{19} \]

Obviously, when \((l,u_\lambda)=1\) the solution of III will also be a solution of II, while the solutions of I and II differ by a factor. Hence we obtain the solutions of problems I and II in the form

\[ u_{\mathrm I}=u_1/d_1;\qquad u_{\mathrm{II}}=-u_1/\sqrt{d_1}. \tag{20} \]

Problems I, II, and III are solved analogously for the periodic case.

The finding of the extremal function can now be reduced to the integration of a partial differential equation. It is convenient to use the apparatus and symbolism of the theory of generalized functions \((^{4-6})\).

By the classical method of the calculus of variations one obtains the equation in variations for the solution of problem III for \(\lambda=1\):

\[ 2D(u_1,\xi)-2\int \xi\,dx-2\sum C_k\xi(x^{(k)})=0, \tag{21} \]

where

\[ D(u_1,\xi)=\int \sum_{|\alpha|=m}D^\alpha u_1 D^\alpha \xi\,dx, \]

\(\xi\) is an admissible variation (any function from \(W_2^{(m)}\) or \(\widetilde W_2^{(m)}\)). We rewrite equation (21) in the form

\[ D(u_1,\xi)=\int_{\Omega}\left[1-\sum_{k=1}^{N}C_k\delta(x-x^{(k)})\right]\xi(x)\,dx \tag{22} \]

(\(\delta(x)\) is the Dirac function).

For the function \(u_1\), by a slight modification of the classical arguments, we obtain the equation

\[ \Delta^m u_1=(-1)^m\left[1-\sum_{k=1}^{N}C_k\delta(x-x^{(k)})\right]; \tag{23} \]

and the boundary conditions

\[ B_k(u_1)\big|_S=0 \tag{24} \]

for the nonperiodic case. In the periodic case there are no boundary conditions.

In the nonperiodic case, from (23) it follows that

\[ u_1=\frac{\Gamma(n/2)2^{-2m}}{\Gamma(n/2+m)\Gamma(m+1)}\,r_k^{2m}- \]

\[ -\sum_{k=1}^{N}C_k \frac{i^{\,n+1}2^{-2m+1}\pi^{n/2+1}}{\Gamma(m+n/2+1)\Gamma(m)} r_k^{\,n-2m} \times \begin{cases} 1, & (n\ \text{odd});\\ \lg r/2\pi i+u_1^*, & (n\ \text{even}), \end{cases} \tag{25} \]

where \(r_k=|x-x^{(k)}|\), \(u_1^*\) is a solution of the polyharmonic equation \(\Delta^m u_1^*=0\), chosen so as to satisfy (24).

In practice, to find \(u_1^*\) one may use the method of integral equations, the grid method, or some other device. It is also convenient to apply some direct method, such as, for example, Ritz’s method, for the direct determination of \(u_1\). Let us also consider the periodic case. For simplicity let \(|\Omega_0|=1\). Denote by \(u^{(k)}\) the periodic solution of the equation

\[ \Delta^m u^{(k)}=(-1)^m\,[1-\delta(x-x^{(k)})];\qquad x\in\Omega_0,\quad x^{(k)}\in\Omega_0. \tag{26} \]

Then

\[ u_1=\sum C_k u^{(k)}. \tag{27} \]

Let \(x\) and \(y\) be coordinate column vectors, and let \(x=Hy\). The periodic function \(\Lambda(x)\) with periods \(H\beta\), equal to \(1-\delta(x-x^{(k)})\) in the parallelepiped \(\Omega_0\), will be a periodic function \(M(y)\) with integer periods \(\beta\). Its value in the fundamental cube will be \(M(y)=[1-\delta(y-y^{(k)})]\), \(-1/2<y\leq 1/2\). This function is expanded in a generalized Fourier series

\[ M(y)=\sum_{|\gamma|\ne 0}\exp[2\pi i(\gamma,y-y^{(k)})], \tag{28} \]

where \(\gamma(\gamma_1,\gamma_2,\ldots,\gamma_n)\) runs through all possible integer values, both positive and negative. Passing to the variables \(x\), we shall have

\[ \Lambda(x)=\sum_{|\gamma|\ne 0}\exp\,[2\pi i(\gamma,H^{-1}(x-x^{(k)}))] =\sum_{|\gamma|\ne 0}\exp\,[2\pi i(\gamma H^{-1},x-x^{(k)})], \tag{29} \]

whence

\[ u^{(k)}=\sum_{|\gamma|\ne 0} \frac{\exp\,[2\pi i(\gamma H^{-1},x-x^{(k)})]}{(\gamma H^{-1})^{2m}}. \tag{30} \]

Formulas (27), (30) make it easy to compute the desired maximum \((l,\varphi)\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
23 XII 1960

References

\(^{1}\) S. L. Sobolev, Matem. sborn., 4 (46), No. 3, 471 (1938).
\(^{2}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^{3}\) S. M. Nikol’skii, Quadrature Formulas, M., 1958.
\(^{4}\) S. L. Sobolev, Matem. sborn., 1 (43), No. 1, 39 (1936).
\(^{5}\) L. Schwarz, Théorie des distributions, 1, Paris, 1950; 2, 1951.
\(^{6}\) I. M. Gel’fand, G. E. Shilov, Generalized Functions and Operations on Them, M., 1958.
\(^{7}\) I. M. Sobol’, DAN, 114, No. 4, 706 (1957).
\(^{8}\) I. M. Sobol’, Application of Expansions in Haar Functions to the Investigation of Integration Grids, Dissertation, M., 1959.
\(^{9}\) I. M. Sobol’, DAN, 132, No. 5, 1041 (1960).
\(^{10}\) N. M. Korobov, DAN, 124, No. 6 (1959).
\(^{11}\) N. M. Korobov, Vestn. MGU, No. 4, 19 (1959).
\(^{12}\) N. S. Bakhvalov, Vestn. MGU, No. 4, 3 (1959).
\(^{13}\) J. H. Halton, Numerische Math., 2, No. 2, 84 (1960).