MATHEMATICS

Academician S. L. SOBOLEV

CUBATURE FORMULAS WITH A REGULAR BOUNDARY LAYER

In one of the preceding notes \((^1)\), error functionals of cubature formulas of the form

\[ l(x)=\mathscr E_{\Omega}(x)-\sum_{j=1}^{N} C_j\delta(x-x^{(j)})=\sum_{\gamma} l_{\gamma}\left(\frac{x}{h}-\gamma\right), \tag{1} \]

were considered, where \(l_{\gamma}(x)\) are functionals with bounded support and bounded norm in \(L_2^{(m)}\), orthogonal to all polynomials of degree \(m\). Such \(l_{\gamma}(x)\), satisfying conditions (10), (11), and (12) of \((^1)\), will be called functionals from \(\mathfrak K(A,L,m)\). Let \(H\) be the matrix of periods of some lattice with determinant equal to one, \(|H|=1\). Representation (1) is, obviously, equivalent to another one, namely:

\[ l(x)=\sum_{\gamma} l_{\gamma}\left(\frac{x}{h}-H\gamma\right). \tag{2} \]

We shall consider formulas for which:

a) The nodes of all \(l_{\gamma}(x)\) are situated at the points \(hH\gamma'\)

\[ l_{\gamma}(x)=\mathscr E_{\gamma}(x)-\sum_{\gamma'} C_{\gamma'}\delta(x-hH\gamma'), \qquad \sum \mathscr E_{\gamma}\left(\frac{x}{h}-\gamma\right)=\mathscr E_{\Omega}(x) \tag{3} \]

b) All \(l_{\gamma}(x)\in\mathfrak K(L,A,s)\).

\[ \tag{4} \]

c) All \(l_{\gamma}(x)\) corresponding to interior points coincide: for \(d(hH\gamma,\Gamma)>2Lh\) we have

\[ l_{\gamma}(x)=l_0(x). \tag{5} \]

Such functionals \(l(x)\) will be called functionals with a regular boundary layer of order \(m\). The aim of our note is to establish the following main theorem:

Theorem. As \(h\to 0\), for all functionals \(l(x)\) with a regular boundary layer of order \(s\), the equality

\[ \|l(x)\|_{L_2^{(m)}}= \left(\frac{h}{2\pi}\right)^m \sqrt{\zeta(H^{-1},2m)}\sqrt{|\Omega|} +O(h^{m+1}) \tag{6} \]

holds.

The proof is based on a number of auxiliary lemmas, which we give here.

Lemma 1. In every functional \(l(x)\) with a regular boundary layer, all coefficients \(C_{\gamma}\) at points whose distance from the boundary is more than \(2Lh\) are equal to \(h^n\).

Indeed:

\[ C_{\gamma}=h^n \sum_{|\gamma'|<L} C_{\gamma-\gamma'}^{(0)}. \tag{7} \]

Since the volume \(\Omega_0\) is equal to unity, \((l_0(x),1)=0\), we have

\[ C_{\gamma}=h^n, \tag{8} \]

which was required to be proved.

By a boundary layer we shall mean the set of nodes \(hH\gamma\) in which \(C_\gamma \ne h^n\). If, in integration, only points interior with respect to \(\Omega\) are used, then the boundary layer will be internal. If, for approximating the functional \(\mathscr E_\Omega(x)\) in \(L_2^{(m)}\), exterior points are also used, then we can obtain a two-sided boundary layer, in which only points lying at a distance from the boundary not greater than \(Lh\) participate, or else an external boundary layer, in which points lying outside \(\Omega\) and at a distance from the boundary not greater than \(2Lh\) participate. Of course, if the function \(\varphi(x)\in L_2^{(m)}(\Omega)\), then only formulas with an internal boundary layer make sense. The number \(2L\) will be called the thickness of the boundary layer.

Let \(m_\gamma(x)=\sum C_{\gamma'}\delta(x-hH\gamma')\). We shall call such functionals point functionals. Let, further, \(m_\gamma(x)\in \mathfrak R(A,L,s)\). Then functionals of the form

\[ m(x)=\sum_{hH\gamma\in B_j} m_\gamma(x-hH\gamma), \tag{9} \]

where \(\gamma\) runs through some set \(B_j\), which is a boundary layer of thickness \(L\), will be called a zero point functional of the boundary layer of order \(s\). A zero point functional of the boundary layer will be external, internal, or two-sided depending on the location of its support. The thickness of this functional, introduced by analogy with the thickness of the boundary layer, is, generally speaking, equal to \(3L\), but it may be smaller. In all that follows it may be regarded as equal to \(2L\).

Lemma 2. The difference of two error functionals \(l^{(1)}(x)\) and \(l^{(2)}(x)\) with regular boundary layers of orders \(s^{(1)}\) and \(s^{(2)}\) is a zero point functional of the boundary layer of order

\[ \min\left(s^{(1)},s^{(2)}\right)-1. \tag{10} \]

The proof of Lemma 2 is based on an auxiliary lemma.

Lemma 3. Let \(m(x)\) be a finite functional of the form

\[ m(x)=\sum_{|H\gamma|<L} C[\gamma]\delta(x-hH\gamma), \tag{11} \]

orthogonal to all polynomials of degree \(s\), \((m(x),x^\alpha)=\sum C[\gamma](hH\gamma)^\alpha=0\) for \(|\alpha|\le s\). The functional \(m(x)\) admits the identical representation:

\[ m(x)=\sum_{j=1}^{n}\left(M_j(x+hHi_j)-M_j(x)\right), \tag{12} \]

where \((M_j(x),x^\alpha)=0\) for \(|\alpha|\le s-1\) and \(S\{M_j(x)\}\) is concentrated in the smallest parallelepiped with edges parallel to the periods \(H\), containing \(S\{m(x)\}\).

The proof of this lemma is carried out by the method of complete induction. We need to establish that the coefficients \(C[\gamma]\) admit the representation

\[ C[\gamma]=\sum_{j=1}^{n}\hat{\Delta}_j C_j[\gamma], \tag{13} \]

where \(\sum C_j[\gamma]\gamma^\alpha=0,\ |\alpha|\le s-1;\ \hat{\Delta}_j\varphi[\gamma]=\varphi[\gamma+i_j]-\varphi[\gamma]\).

Let us show that

\[ C[\gamma_1,\gamma_2,\ldots,\gamma_n] = C[\gamma_1,\gamma_2,\ldots,\gamma_{n-1},\gamma_n+1] - C[\gamma_1,\gamma_2,\ldots,\gamma_n] + C^*[\gamma_1,\gamma_2,\ldots,\gamma_{n-1}], \tag{14} \]

where \(C^*[\gamma_1,\gamma_2,\ldots,\gamma_{n-1}]\) is again orthogonal to all polynomials of degree \(s\), while \(C[\gamma_1,\gamma_2,\ldots,\gamma_n]\) is orthogonal to polynomials of degree \(s-1\), and its support is contained in the smallest parallelepiped containing the support of \(C_n\). Hence Lemma 3 will follow. As \(C[\gamma_1,\gamma_2,\ldots,\gamma_n]\) and \(C^*[\gamma_1,\gamma_2,\ldots,\gamma_{n-1}]\) it suffices to take the expression

\[ C^*[\gamma_1,\gamma_2,\ldots,\gamma_{n-1}] = \sum_{\gamma_n'=-L}^{L} C[\gamma_1,\gamma_2,\ldots,\gamma_{n-1},\gamma_n']; \tag{15} \]

\[ C_n[\gamma_1,\gamma_2,\ldots,\gamma_n]= \begin{cases} 0, & |\gamma_n|\ge L,\\[6pt] \displaystyle \sum_{\gamma_n'=-L}^{\gamma_n-1} C[\gamma_1,\gamma_2,\ldots,\gamma_{n-1},\gamma_n'] - (\gamma_n+L)\,C^*(\gamma_1,\gamma_2,\ldots,\gamma_{n-1}). \end{cases} \tag{16} \]

Verification of formula (14) and the orthogonality of \(C^*[\gamma_1,\gamma_2,\ldots,\gamma_{n-1}]\) to polynomials of degree \(s\) are obvious. The orthogonality of \(C_n[\gamma_1,\gamma_2,\ldots,\gamma_n]\) to all polynomials of degree \(s-1\) follows from the well-known summation-by-parts formula:

\[ \sum_{\gamma}[\varphi(\gamma+1)-\varphi(\gamma)]\psi(\gamma) = \sum_{\gamma}\varphi(\gamma)\bigl(\psi(\gamma)-\psi(\gamma-1)\bigr). \tag{17} \]

It is enough to note that \(x_n^{\alpha_n}\) can be represented in the form

\[ x_n^{\alpha_n}=\frac{1}{\alpha_n}\,\hat{\Delta}_n B_{\alpha_n}(x), \]

where \(B_{\alpha_n}\) is a Bernoulli polynomial, and to use (16).

Lemma 3 could also be proved in another way—by passing to the Fourier image. Its dual formulation is as follows:

Lemma 3a. Let \(Z\) be the class of rational functions of the form \(\Psi(z)=P(z)/z^k\), where \(P(z)\) is a polynomial in the variable \(z(z_1,\ldots,z_n)\), and \(z^k=z_1^{k_1}\ldots z_n^{k_n}\). Every function \(\varphi(z)\) having at the point \(I(1,1,\ldots,1)\) a zero of multiplicity \(m\) is representable in the form

\[ \varphi(z)=\sum_{j=1}^{n}(z_j-1)\varphi_j(z), \tag{18} \]

where \(\varphi_j(z)\) are of the same class \(Z\) and have at the point \(I\) zeros of multiplicity \((m-1)\), and, moreover, the degree of the polynomial \(P_j(z)\) in each variable \(z_j\) is no higher than the degree of \(P(z)\) and \(k_j'\le k\).

Apparently, the proof in the text is no longer than a possible proof of this dual theorem, especially if one takes into account the establishment of their equivalence.

Lemma 3a and Lemma 3 are a particular example of lemmas on the representation of analytic functions having, at a given point \(z^{(0)}\), a zero of multiplicity \(m\), in the form

\[ \varphi(z)=\sum (z_j-z^{(0)})\varphi_j(z) \tag{19} \]

and of dual lemmas on the corresponding representation of generalized functions \(\psi\in K^{(s)}\), where \((\psi(x)*x^\alpha)=0,\ |\alpha|\le s\).

For example, L. Schwartz’s theorem on the representation of any generalized function in the form of a differential operator applied to a continuous function is, in essence, also a lemma of the type of Lemma 3.

Corollary. The sum

\[ \sum_{hH\gamma\in\Omega} m_0(x-hH\gamma)=M_0(x), \tag{20} \]

where \(m_0(x)\) are zero point functionals from \(\mathfrak K(A,L,s+1)\), constitute a functional of a zero boundary layer of order \(s\).

This corollary is obtained immediately if in the left-hand side of (18) one replaces \(m_0\) by its expression on the basis of Lemma 3 and interchanges the order of summation. Lemma 2 follows immediately from Lemma 3.

Corollary of Lemma 2. Every functional \(l(x)\) with a boundary layer of order \(s \geqslant m\) can be represented in the form

\[ l(x)=\sum_{hH\gamma\in\Omega} l^*\left(\frac{x}{h}-H\gamma\right) +\sum_{\gamma\in B} l_\gamma^{**}\left(\frac{x}{h}-H\gamma\right), \tag{21} \]

where \(B\) is a boundary layer of thickness \(L\), and moreover in such a way that

\[ l^*\left(\frac{x}{h}-H\gamma\right)\in \mathfrak K(A,L,s), \tag{22} \]

where \(s_1\) is any number greater than \(s\).

It suffices to represent, in the form (9), the difference between our functional and an arbitrary error functional with a regular boundary layer of order \(s_1\). Consider a linear functional \(l(x)\) with a regular boundary layer of order \(s \geqslant m\). The representation

\[ l(x)=1-\Phi_0(h^{-1}H^{-1}x)-l^{(1)}(x), \tag{23} \]

is valid, where \(l^{(1)}(x)\) is a functional with a regular exterior boundary layer for the domain \(\overline{\Omega}=E_n\setminus\Omega\). The extremal function for the functional \(l(x)\) has the form

\[ u(x)=l(x)*G(x) \tag{24} \]

(see (1)). Using (23), we obtain:

\[ u(x)=u_0(x)+C-\bigl(l^{(1)}(x)*G(x)\bigr), \tag{25} \]

where \(u_0(x)\) is an elementary solution of the periodic extremal problem. Let us write explicitly the expression for the norm of the functional \(l(x)\):

\[ \|l(x)\|^2=(l(x),u(x))=(l(x),u_0(x)) -\left.(l(x)*G(x)*l^{(1)}(-x))\right|_{x=0} \tag{26} \]

and replace \(l(x)\) and \(l^{(1)}(x)\) by their representation (21). We obtain, repeating the estimates given in (1):

\[ \left. l(x)*G(x)*l^{(1)}(-x)\right|_{x=0}=O(h^{2m+1}). \tag{27} \]

By direct calculation one can also show that

\[ (l(x),u_0(x))=\frac{h^{2m}}{(2\pi)^{2m}}\zeta(H^{-1},2m)|\Omega|+O(h^{2m+1}). \tag{28} \]

Comparing (25), (26), and (27), we obtain the proof of the main theorem.

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

Received
3 V 1965

CITED LITERATURE

1. S. L. Sobolev, Dokl. Akad. Nauk SSSR, 162, No. 5 (1965).