UDC 518-517.392

MATHEMATICS

V. I. Polovinkin

WEIGHTED CUBATURE FORMULAS

(Presented by Academician S. L. Sobolev on 18 V 1967)

In the works of S. L. Sobolev \((^{1-5})\), cubature formulas were studied for a bounded domain of integration with constant weight and nodes at the points of regular lattices. In these works an algorithm was indicated for constructing sequences of cubature formulas that are asymptotically optimal in the class \(L_2^{(m)}(E)\). For the error functionals of optimal cubature formulas, asymptotically unimprovable estimates of norms in \(L_2^{(m)*}(E_n)\) were obtained. In the present note the main results of \((^{1-5})\) are generalized to the case in which the weight function is integrable in the square.

We introduce the following notation: \(L_2^{(m)}(E_n)\) is the space of classes of functions possessing, in the \(n\)-dimensional space \(E_n\), generalized derivatives integrable in the square up to order \(m\) (we shall assume that \(2m>n\)) and with norm

\[ \|f\|_{L_2^{(m)}(E_n)}=\left\{\int_{E_n}\sum_{|\alpha|=m}(D^\alpha f)^2\,dx\right\}^{1/2} \tag{1} \]

(for an integral vector \(\alpha\), \(|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_n\), \(D^\alpha=\partial^{|\alpha|}/\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}\)); \(\Omega\) is a domain in \(E_n\) with piecewise smooth boundary \(\Gamma\); \(H\) is a square matrix of order \(n\times n\); \(\Omega_0\) is a fundamental domain for the matrix \(H\), i.e., such that the characteristic function \(\mathscr E_{\Omega_0}\) of the domain \(\Omega_0\) satisfies, for all \(x\), the identity

\[ \sum_\gamma \mathscr E_{\Omega_0}(x-H\gamma)\equiv 1; \tag{2} \]

\(\Omega_{h,\gamma}\) are domains such that \(\mathscr E_{\Omega_{h,\gamma}}(x)\equiv \mathscr E_{\Omega_0}(x/h-H\gamma)\); \(h\) is a numerical parameter taking positive values.

Definition 1. A set of interpolation operators \(\{I^h\}\) is called a family of uniformly distributed operators if all the operators of \(\{I^h\}\) are defined on \(L_2^{(m)}(E_n)\), i.e., they assign to each function \(f,\ f\in L_2^{(m)}(E_n)\), functions \(I^h f\), and are representable in the form of sums of operators \(I_\gamma^h\),

\[ I^h=\sum_\gamma I_\gamma^h, \tag{3} \]

possessing the following properties:

a)

\[ (I_\gamma^h f)(x)=\sum_{|\gamma'|\le L} g_{\gamma,\gamma+\gamma'}^h(x) f\bigl(Hh(\gamma+\gamma')\bigr), \tag{4} \]

where \(g_{\gamma,\gamma+\gamma'}^h(x)\) are functions equal to zero outside \(\Omega_{h,\gamma}\); \(L\) is a constant independent of \(h\) and \(\gamma\);

b)

\[ \left|g_{\gamma,\gamma+\gamma'}^h(x)\right|\le M, \tag{5} \]

\(M\) is a constant independent of \(h,\gamma\), and \(\gamma'\);

c)

\[ I_\gamma^h\left(x_1^{\alpha_1},x_2^{\alpha_2}\cdots x_n^{\alpha_n}\right) = \mathscr E_{\Omega_{h,\gamma}} x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n} \quad \text{for } |\alpha|\le m. \tag{6} \]

Theorem 1. For every family of uniformly distributed operators \(\{I^h\}\) there exists a number \(A\) such that, for any weight function \(g\) square-integrable in \(\Omega\), for the norms in \(L_2^{(m)*}(E_n)\) of the functionals of the errors of the cubature formulas

\[ \int_\Omega gf\,dx \cong \int_\Omega g I^h f\,dx \tag{7} \]

\(l_g^h\), the estimates

\[ \|l_g^h\|_{L_2^{(m)*}(E_n)} \leq A\left[\int_\Omega |g|^2\,dx\right]^{1/2} h^m = A\|g\|_{L_2(\Omega)}h^m \tag{8} \]

hold.

We outline the proof of Theorem 1. Define the functionals \(l_g^{h,\gamma}\) by the equalities

\[ l_g^{h,\gamma}(f)=\int_{\Omega\cap \Omega_{h,\gamma}} g[f-I^h f]\,dx . \tag{9} \]

Analogously to the proof of equality (24) from \((^1)\), the validity of the formula

\[ \|l_g^h\|_{L_2^{(m)*}(E_n)}^2 = \sum_{\gamma_1}\sum_{\gamma_2} \left[ G(x)*l_g^{h,\gamma_1}(x)*l_g^{h,\gamma_2}(-x) \right]_0 \tag{10} \]

is shown.

Here \(G(x)\) in (10) is the fundamental solution of the polyharmonic equation

\[ \Delta^m u=0 . \tag{11} \]

Lemma 1. The estimate

\[ \left|G(x)*l_g^{h,\gamma_1}(x)*l_g^{h,\gamma_2}(-x)\right| \leq Kh^{2m} \frac{\|g\|_{L_2(\Omega_{h,\gamma_1})}\|g\|_{L_2(\Omega_{h,\gamma_2})}} {\left[1+|H(\gamma_1-\gamma_2)|^2\right]^{n/2+1}} . \tag{12} \]

holds.

The proof of Lemma 1 is analogous to the proof of Lemma 2 from \((^1)\). Substituting (12) into (10) and applying Theorem 275 from \((^6)\), p. 239, we obtain the assertion of Theorem 1.

Definition 2. A set of interpolation operators \(\{I^h\}\) is called a family of interpolation operators with a regular boundary layer if \(\{I^h\}\) is a uniformly distributed family and there exists a constant \(T\) characterized by the following property: if the integer vectors \(\gamma_1,\gamma_2\) are such that the distances from the points \(Hh\gamma_1, Hh\gamma_2\) to \(\Gamma\) are greater than \(Th\), then the functions \(g_{\gamma_1,\gamma_1+\gamma}^{h}(x)\), \(g_{\gamma_2,\gamma_2+\gamma}^{h}(x)\), for all \(\gamma\), \(|\gamma|\leq L\), satisfy the identities

\[ g_{\gamma_1,\gamma_1+\gamma}^{h}(x) \equiv g_{\gamma_2,\gamma_2+\gamma}^{h}(x-hH\gamma_1+hH\gamma_2). \tag{13} \]

From the definition of functionals with a regular boundary layer it follows that

Lemma 2. If the family of interpolation operators \(\{I^h\}\) is uniformly distributed, then the set of functionals \(\{l_{\xi}^{h}\}\) is a set of functionals with a regular boundary layer satisfying the conditions of the main theorem from \((^2)\).

Theorem 2. If \(\{I^h\}\) is a family of interpolation operators with a regular boundary layer, and \(g\) is a weight function of class \(L_2(\Omega)\), then for the norms in \(L_2^{(m)*}(E_n)\) of the functionals \(l_g^h\), as \(h\to 0\), the estimate

\[ \|l_g^h\|_{L_2^{(m)*}(E_n)} = (2\pi)^{-m}\|g\|_{L_2(\Omega)}h^m \sqrt{\xi(H^{-1},2m)}+o(h^m) \tag{14} \]

holds.

Theorem 2 is first proved for the case of a piecewise constant weight, and then, with the help of Theorem 1, is extended to the general case. At the first stage of the proof, Lemma 2, the main theorem from \((^2)\), and representation (10) for the norm of \(l_g^h\) in \(L_2^{(m)*}(E_n)\) are used essentially.

Definition 3. A cubature formula is said to be optimal in \(L_2^{(m)*}(E_n)\) if the norm of its error functional \(l_{g,0}^h\) is minimal in \(L_2^{(m)*}(E_n)\) among the norms of all functionals of the form:

\[ l=g\mathcal E_{\Omega}-\sum_{\substack{\gamma\\ |hH\gamma|\in\Omega}} c_\gamma\delta(x-hH\gamma),\qquad l\left(x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}\right)=0 \quad \text{for } |\alpha|\le m-1, \tag{15} \]

where \(c_\gamma\) are constant coefficients.

Theorem 3. If \(\{I^h\}\) is a family of interpolation operators with a regular boundary layer, and \(g\) is a function in \(L_2(\Omega)\), then as \(h\to 0\) the estimate

\[ \left\|l_g^h\right\|_{L_2^{(m)*}(E_n)} - \left\|l_{g,0}^h\right\|_{L_2^{(m)*}(E_n)} = o(h^m) \tag{16} \]

holds.

The proof of Theorem 3 is based on approximating the weight function \(g\) by piecewise-constant functions. The proof essentially uses Theorem 1, Theorems 5 and 6 from \((^3)\), and the following lemma, whose validity follows from the Hilbert nature of the spaces \(L_2^{(m)}(E_n)\) and \(L_2^{(m)*}(E_n)\).

Lemma 3. For weight functions \(g_1\) and \(g_2\) integrable in \(\Omega\), the optimal cubature formulas have error functionals \(l_{g_1,0}^h\) and \(l_{g_2,0}^h\) satisfying the equality

\[ l_{g_1,0}^h+l_{g_2,0}^h=l_{g_1+g_2,0}^h. \tag{17} \]

The author expresses gratitude to S. L. Sobolev for his assistance in the work.

Novosibirsk State
University

Received
12 V 1967

REFERENCES

\(^{1}\) S. L. Sobolev, DAN, 162, No. 5 (1965).
\(^{2}\) S. L. Sobolev, DAN, 163, No. 3 (1965).
\(^{3}\) S. L. Sobolev, DAN, 164, No. 2 (1965).
\(^{4}\) S. L. Sobolev, Lectures on the Theory of Cubature Formulas, Part 1, Novosibirsk, 1964.
\(^{5}\) S. L. Sobolev, Lectures on the Theory of Cubature Formulas, Part 2, Novosibirsk, 1965.
\(^{6}\) G. Hardy, J. E. Littlewood, G. Polya, Inequalities, IL, 1948.