Reports of the Academy of Sciences of the USSR

1. Volume 164, No. 2

MATHEMATICS

Academician S. L. SOBOLEV

OPTIMAL FORMULAS OF MECHANICAL CUBATURES WITH NODES AT POINTS OF REGULAR LATTICES

In previous notes \((^{1-4})\) we investigated the norm of the error functional in \(L_2^{(m)*}\) for cubature formulas for finite functions with constant coefficients and nodes situated in a regular lattice, and also for formulas with a regular boundary layer in domains with a sufficiently smooth boundary. In the present note we establish that the optimal coefficients of cubature formulas with a regular lattice of nodes have the same principal term in the norm of the error functional as do formulas with a regular boundary layer.

By Babushka’s theorem \((^6)\), the optimal coefficients of cubature formulas with error functional

\[ l(x)=\mathcal{E}_{\Omega}(x)-\sum_{\beta} C_{\beta}\delta(x-hH\beta) \]

are characterized by the fact that the solutions from \(L_2^{(m)}\) of the equation \(\Delta^m u_0=(-1)^m l_0(x)\) take at the points \(hH\gamma\) values coinciding with the values of a certain polynomial of degree \(m-1\). In other words,

\[ u_0(hH\beta)=\left.(l_0(x)*G(x))\right|_{x=hH\beta}=P(hH\beta). \]

Let us also consider a cubature formula with the same nodes and with a regular boundary layer of order \(2m+2\)

\[ l_1(x)=\sum_{\beta'} l_{\beta'}^{(1)}\left(\frac{x}{h}-H\beta'\right) =\mathcal{E}_{\Omega}(x)-\sum_{\beta} C_{\beta}\delta(x-hH\beta). \]

The square of the norm of the error of the cubature formula is a second-degree polynomial in the coefficients \(C_\beta\)

\[ \|l(x)\|_{L_2^{(m)*}}^2=(l(x)*G(x)*l(-x))\big|_{x=0}=\Psi(C)= \]

\[ =(\mathcal{E}_{\Omega}(x)*G(x)*\mathcal{E}_{\Omega}(-x))\big|_{x=0} -2\sum_{\beta} C_{\beta}(\mathcal{E}_{\Omega}(x)*G(x-hH\beta))\big|_{x=0} + \]

\[ +\sum_{\beta}\sum_{\beta'}G(hH(\beta-\beta'))C_{\beta}C_{\beta'}. \]

The difference \(\Psi(C)-\Psi(C^{(0)})\) is, consequently, in its turn a second-degree polynomial in the differences \(C_\beta-C_\beta^{(0)}\). It can be proved that this difference is expressed simply by the quadratic form

\[ \Psi(C)-\Psi(C^{(0)})= \sum_{\beta}\sum_{\beta'}G(hH(\beta-\beta')) (C_{\beta}-C_{\beta}^{(0)})(C_{\beta'}-C_{\beta'}^{(0)}). \]

Denoting

\[ \rho[\beta]=0,\quad hH\beta\notin\Omega;\qquad \rho[\beta]=C_{\beta}-C_{\beta}^{(0)},\quad hH\beta\in\Omega, \tag{1} \]

we obtain the form already studied by us earlier in note \((^5)\):

\[ \Psi(C)-\Psi(C^{(0)})=\Delta_{hH}(U_{hH}[\beta],U_{hH}[\beta]), \]

where

\[ U_{hH}[\beta]=\rho[\beta]*G_{hH}[\beta] \]

and associated with the difference generalization of the polyharmonic potential.

Theorem 1. The values of the difference potential \(\rho[\beta] * G_{hH}[\beta] = U_{hH}[\beta]\) at interior points of the domain \(\Omega\) differ by a polynomial of degree \(m-1\) from the values \(u(hH\beta)\), where

\[ u(x)=l(x)*G(x). \tag{2} \]

Proof. For the difference \(u(x)-u_0(x)\) we have \(u(x)-u_0(x)=(l(x)-l_0(x))*G(x)=\sum (C_\beta-C_\beta^{(0)})\delta(x-hH\beta)*G(x)\), and, hence, \(u(hH\beta)-u_0(hH\beta)=\rho[\beta]*G_{hH}[\beta]\), whence from (1) it follows that \(U_{hH}[\beta]-u(hH\beta)=-u_0(hH\beta)=-P(hH\beta)\) for \(hH\beta\in\Omega\). Theorem 1 is proved.

Theorem 2. The potential \(U_{hH}[\beta]\) gives the absolute minimum of the functional \(\Delta_{hH}(\varphi[\beta],\varphi[\beta])\) among all functions \(\varphi[\beta]\) belonging to \(l_2^{(m)}\) and coinciding at the points \(hH\beta\in\Omega\) with the values \(U_{hH}[\beta]\),

\[ \Delta_{hH}(U_{hH}[\beta],\,U_{hH}[\beta])\leq \Delta_{hH}(\varphi[\beta],\,\varphi[\beta]), \]

if \(\varphi[\beta]=U_{hH}[\beta]\) for \(hH\beta\in\Omega\).

The proof is based on several lemmas. First it is necessary to extend the scalar products \(\Delta_{hH}\) to a somewhat broader space, and then to use Green’s formula for its transformation. Let \(M_p^{(l)}(N)\) be the space of functions \(\varphi(x)\) defined in \(E_n\) with norm determined by the equality:

\[ \|\varphi\|_{M_p^{(l)}(N)} = \left\{ \int_{-\infty}^{+\infty} \left[ \max_{|y_j-z_j|<N} \sum_{|\alpha|=m}(D^\alpha\varphi(y))^2 \right]^{p/2} dz \right\}^{1/p}. \]

Lemma 1. The spaces \(M_p^{(l)}(N)\) for different values are equivalent to one another.

The proof is based on two inequalities:

\[ \|\varphi\|_{M_p^{(l)}(N_1)}^p \geq \|\varphi\|_{M_p^{(l)}(N_2)}^p \quad\text{for } N_1\geq N_2; \tag{3} \]

\[ \|\varphi\|_{M_p^{(l)}((2k+1)N)}^p \leq (2k+1)^n\|\varphi\|_{M_p^{(l)}(N)}^p, \tag{4} \]

which are established elementarily. From (3) and (4) the lemma follows almost immediately.

Lemma 2. Let \(\varphi[\beta]\in l\), i.e., let it have differences of order \(l\), summable with degree \(p\) over the whole space. Then there exists a function \(\varphi(x)\in M_p^{(l)}(N)\), coinciding with \(\varphi[\beta]\) at all points \(x=hH\beta\), \(\varphi(hH\beta)=\varphi[\beta]\), and such that

\[ \|\varphi(x)\|_{M_p^{(l)}(N)} \leq K\|\varphi[\beta]\|_{l_p^{(l)}}. \]

The proof does not differ from the analogous assertion for \(L_p^{(l)}\). V. S. Ryabenkii and A. F. Filippov [7] constructed an interpolation operator \(\Pi\varphi[\beta]\) which assigns to any function \(\varphi[\beta]\) given on the integer lattice \(\beta\) a certain function \(\varphi(x)\) possessing the properties: 1) \(\varphi(\beta)=\varphi[\beta]\), 2) \(|D^\alpha\varphi|\leq K\max_{|hN\beta-x|<Lh}\Delta^\alpha\varphi[\beta]\). The analysis shows that the Ryabenkii–Filippov operator gives the interpolation function needed by us.

Lemma 3. The values of any function \(\varphi(x)\in M_p^{(l)}(N)\) at the points \(hH\beta\): \(\varphi[\beta]=\varphi(hH\beta)\), constitute an element of the space \(l_p^{(l)}\).

The proof of Lemma 3 is carried out by means of estimates and is based on the integral representation of differences through derivatives.

Theorem 3. In the space \(M_p^{(l)}\), finite functions form a dense set.

The proof of this theorem is rather laborious, but it differs little from the proof of the theorem on the density of finite functions in \(L\) (see \((^{8,9})\)), and therefore we shall not reproduce the corresponding line of reasoning here.

Theorem 4. In the space \(l_p^{(l)}\), the finite functions form a dense set.

The proof of Theorem 4 follows from Theorem 3, and also from the equivalence of \(l_p^{(l)}\) and \(M_p^{(l)}\), when a correspondence is established between them by the methods indicated in Lemmas 2 and 3.

Lemma 4 (Green’s formula). The formula

\[ \Delta_{hH}(u[\beta],\,v[\beta])=(u[\beta],\,L_{hH}[\beta]*v[\beta]), \tag{5} \]

which serves as the definition of the scalar product \(\Delta_{hH}\), in the case where both functions \(u[\beta]\), \(v[\beta]\) decrease exponentially at infinity, remains valid if one of the functions \(u[\beta]\) or \(v[\beta]\) is an arbitrary element of \(l_2^{(m)}\), and the other is finite.

The proof of this lemma follows from Theorem 4 and the equivalence of the scalar product \(\Delta_{hH}(\varphi,\psi)\) to the square of the norm of the function \(u[\beta]\) in \(l_2^{(m)}\) (see (5)).

Lemma 5. If the density \(\rho[\beta]\) is orthogonal to all polynomials of degree \(m-1\), \((\rho[\beta],(hH\beta)^\alpha)=0\), \(|\alpha|\leq m-1\), then the function \(U_{hH}[\beta]=\rho[\beta]*G_{hH}[\beta]\) is an element of \(l_2^{(m)}\).

The proof is carried out in the same way as the corresponding proof for the convolution \(l(x)*G(x)\), given in \((^{1,10})\).

We now indicate the method of proving Theorem 2. Let \(u[\beta]=0\) for \(hH\beta\in\Omega\) and \(L_{hH}*v[\beta]=0\) for \(hH\beta\notin\Omega\), with \(u\in l_2^{(m)}\) and \(v\in l_2^{(m)}\). Then \(\Delta_{hH}(u[\beta],v[\beta])=0\), as follows easily from (5).

Put \(\varphi[\beta]-U[\beta]=u[\beta]\). Formula (5) is valid for the product \((U[\beta],u[\beta])\), since \(u[\beta]=0\) for \(hH\beta\in\Omega\), by assumption, and \(L_{hH}[\beta]*U[\beta]=0;\ hH\beta\notin\Omega\), because the operator \(L_{hH}[\beta]\) is inverse to convolution with \(G_{hH}[\beta]\).

Using Lemma 5, we shall have

\[ \begin{aligned} \Delta_{hH}(\varphi[\beta],\varphi[\beta]) &=\Delta_{hH}(\varphi[\beta]-U[\beta],\varphi[\beta]-U[\beta]) \\ &\quad +2\Delta_{hH}(\varphi[\beta]-U[\beta],U[\beta]) +\Delta_{hH}(U[\beta],U[\beta])= \\ &=\Delta_{hH}(\varphi[\beta]-U[\beta],\varphi[\beta]-U[\beta]) +\Delta_{hH}(U[\beta],U[\beta]), \end{aligned} \]

whence Theorem 5 follows at once.

Theorem 5. The deviation of the square of the norm of the error functional \(l(x)\) from the minimal one satisfies the inequality

\[ \|l(x)\|^2-\|l_0(x)\|^2\leq \Delta_{hH}(u(hH\beta),u(hH\beta)), \tag{6} \]

where \(u(x)\) is defined by formula (2).

This theorem follows from Theorems 1 and 2.

In order to complete our estimate, it remains to prove the last theorem.

Theorem 6. The inequality

\[ \Delta_{hH}(u(hH\beta),u(hH\beta))\leq Kh^{2m+1} \tag{7} \]

is valid.

Before giving the idea of the proof, let us note that from Theorem 6 the result we need will follow. Indeed, for the functional \(l(x)\) with a regular boundary layer the formula is valid (see \((^{2})\))

\[ \|l(x)\|_{L_2^{(m)*}}^2 =(h/2\pi)^{2m}\zeta(H^{-1}\mid 2m)|\Omega|+O(h^{2m+1}). \tag{8} \]

Comparing (8), (7), and (6), we see that for \(l_0(x)\) the same estimate is valid:

\[ \|l_0(x)\|_{L_2^{(m)*}}^2 =(h/2\pi)^{2m}\zeta(H^{-1}\mid 2m)|\Omega|+O(h^{2m+1}). \]

Hence

\[ \|l_0(x)\|_{L_2^{(m)*}} =(h/2\pi)^m\sqrt{\zeta(H^{-1}\mid 2m)}\,\sqrt{|\Omega|} +O(h^{m+1}). \]

The proof of Theorem 6 is based on the representation \(u(x)=u_0(x)-w(x)\), where \(u_0(x)\) is the periodic solution of the equation \(\Delta^m u=(-1)^m(1-\Phi_0(h^{-1}H^{-1}x))\), given in (2). Here \(w(x)=G(x)*l_2(x)\), and \(l_2(x)\) is the error functional with a regular boundary layer for the exterior of the domain \(\Omega\):

\[ l_2(x)=1-\Phi_0(h^{-1}H^{-1}x)-l_1(x) =\sum_{\beta\in\Omega} l_\beta^{(2)}\!\left(\frac{x}{h}-H\beta\right), \]

where \((l_\beta^{(2)}(x),x^\alpha)=0\) for \(|\alpha|<2m+2\), and all \(l_\beta^{(2)}\) for \(d(hH\beta,\Omega)>Lh\) are equal to \(l_0^{(2)}(x)\).

Let \(u_{\beta_1}^{(j)}(x)=G(x)*l_{\beta_1}^{(j)}(x)\), \(j=1,2\). Computing \(\Delta_{hH}(u(hH\beta),u(hH\beta))\), we obtain

\[ \Delta_{hH}(u(hH\beta),u(hH\beta))= \]

\[ =(u_1(hH\beta),L_{hH}[\beta]*u_0(hH\beta))-(w(hH\beta),u_1(hH\beta)). \tag{9} \]

The first term on the right-hand side of (9) is equal to zero. Indeed, \(u_0(hH\beta)\), obviously, is constant, and the convolution operator with \(L_{hH}[\beta]\) is orthogonal to all polynomials of degree below \(m\). We transform the second term into a sum, similarly to what was done in (4):

\[ \Delta_{hH}(w(hH\beta),u_1(hH\beta)) =\sum_{\beta_1,\beta_2}\Delta_{hH}\bigl(u_{\beta_1}^{(1)}(hH\beta),u_{\beta_2}^{(2)}(hH\beta)\bigr). \tag{10} \]

Expanding further the right-hand side of (10),

\[ \Delta_{hH}\bigl(u_{\beta_1}^{(1)}(hH\beta),u_{\beta_2}^{(2)}(hH\beta)\bigr) =\left(u_{\beta_1}^{(1)}(-hH\beta)*L_{hH}[\beta]*(G(x)*l_{\beta_2}^{(2)}(x))\big|_{x=hH\beta}\right), \]

we obtain

\[ v_{\beta_2}[\beta]=L_{hH}[\beta]*\bigl[(G(x)*l_{\beta_2}^{(2)}(x))\big|_{x=hH\beta}\bigr]= \]

\[ =\sum_{\beta'}L_{hH}(\beta-\beta')\int G(hH\beta'-x)\, l_{\beta_2}^{(2)}\!\left(\frac{x}{h}\right)\,dx= \]

\[ =\int\left[\sum_{\beta'}L_{hH}(\beta-\beta')G(hH\beta'-x)\right] l_{\beta_2}^{(2)}\!\left(\frac{x}{h}\right)\,dx. \]

The function

\[ \sum_{\beta'}L_{hH}(\beta-\beta')G(hH\beta'-x)=\tau(x), \]

as was shown in (5), decreases exponentially at infinity. Hence \(v_{\beta_2}[\beta]\le e^{-\eta|\beta|}\). Further, by assumption, \(u_{\beta_1}^{(1)}[\beta]\) decreases and \(u_{\beta_1}^{(1)}(x)\le Kh^{2m+2n+2}/(h^2+x^2)^{(n+1)/2}\). Thus

\[ \Delta_{hH}(u_{\beta_1}[\beta],v_\beta[\beta]) \le \frac{Kh^{2m+2n+2}}{(h^2+x^2)^{(n+1)/2}}. \]

An integral estimate of the sum

\[ \sum_{\beta_1,\beta_2}\Delta_{hH}(u_{\beta_1}[\beta],v_{\beta_2}[\beta]), \]

similar to that carried out in (5), gives the proof of Theorem 5.

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

Received
24 V 1965

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