UDC 517.392

MATHEMATICS

V. I. POLOVINKIN

CUBATURE FORMULAS IN (L_2^{(m)}(\Omega))

(Presented by Academician S. L. Sobolev, May 28, 1969)

In ((^{1-5})) Sobolev investigated formulas of approximate integration in the classes (L_2^{(m)}(E_n)), (L_2^{(m)}(\Omega)). The case (L_2^{(m)}(E_n)) was studied most fully. In particular, cubature formulas with a regular boundary layer and with error functionals were determined ((^{2,4}))

[
l^h \in L_2^{(m)*}(E_n),\quad 2m > n,
]

[
(l^h,f)=\int_\Omega pf\,dx-\sum_{hH\gamma\in\Omega} c_\gamma^h f(hH\gamma):\ \int |p|^2\,dx=|p|_{L_2(\Omega)}^2<\infty,
\tag{1}
]

where (p(x)\equiv 1), and such that, as (h\to 0) and for given (p),

[
|l^h|{B^*}^2-B(H)|p|{L_2(\Omega)}^2h^{2m}=o(h^{2m}):\quad
B,}(H)=(2\pi)^{-2m}\sum_{\beta}'|\beta H^{-1}|^{-2m
\tag{2}
]

if (B=L_2^{(m)}(E_n)). It was shown ((^{2,5})) (see also ((^7))) that for any ({l^h}) of the form (1) and (B=L_2^{(m)}(E_n)), the left-hand side of (2) is greater than some (\alpha(h)), (\alpha(h)=O(h^{2m+1})) as (h\to 0), i.e., that families of formulas with a regular boundary layer are asymptotically optimal on lattices in (L_2^{(m)}(E_n)), or, in other words, the asymptotically optimal families of error functionals of these formulas. In ((^6)) the indicated results were generalized to the case (p\in L_2(\Omega)), and it was shown that for such weights the functionals (l^h), forming an asymptotically optimal family, satisfy (2) with (B=L_2^{(m)}(E_n)). The methods of proving the assertions in ((^{1-6})) were based on the theory of the polyharmonic equation (\Delta^m u=0), which is the Euler equation for the functionals (1) in (L_2^{(m)}(\Omega)), (L_2^{(m)}(E_n)). The case of the spaces (L_2^{(m)}(E_n)) turned out to be simpler than the case (L_2^{(m)}(\Omega)), since in the study of formulas of approximate integration in (L_2^{(m)}(E_n)) no boundary-value problems arise for the polyharmonic equation ((^{1,3})). In the present paper the results from ((^{1-6})) concerning asymptotically optimal families of cubature formulas in (L_2^{(m)}(E_n)) are generalized to (L_2^{(m)}(\Omega)) by means of methods of the theory of Hilbert spaces.

Notation: (\Omega) is a domain in the (n)-dimensional Euclidean space (E_n) with a sufficiently “good” boundary (for example, piecewise smooth); (\gamma) are integer vectors from (E_n); (H) is an (n\times n) matrix, (|H|=1); (L_2=L_2(\Omega)); (h) is a positive parameter; (m) is an integer greater than (n/2); (L_2^{(m)}(E_n)), (L_2^{(m)}(\Omega)) are known Banach spaces of classes of functions that may be regarded as Hilbert spaces ((^1)), if the scalar product in them is defined as follows: let (F,G\in L_2^{(m)}(M)) ((M) is one of the sets (E_n,\Omega)); (f,g) are functions, (f\in F), (g\in G); then

[
(F,G){L_2^{(m)}(M)}
=
\int_M
\left(
\sum}^{n}\cdots\sum_{j_m=1}^{n
\frac{\partial^m f}{\partial x_{j_1}\cdots \partial x_{j_m}}
\frac{\partial^m g}{\partial x_{j_1}\cdots \partial x_{j_m}}
\right)\,dx.
]

(\dot L_2^{(m)}(\Omega,E_n)), (\dot L_2^{(m)}(E_n\setminus\Omega,E_n)) are subspaces of (L_2^{(m)}(E_n)), defined as follows:
(F \in \dot L_2^{(m)}(M,E_n)) ((M) is one of the sets (\Omega, E_n\setminus\Omega)) if and only if there is a function (f) in (F) that is zero in (E_n\setminus M); (\Pi) is the operator assigning to (F\in L_2^{(m)}(E_n)) the class (\Pi F\in L_2^{(m)}(\Omega)) consisting of the traces on (\Omega) of functions from (F); (\dot L_2^{(m)}(\Omega)) is the subspace of (L_2^{(m)}(\Omega)) consisting of images of elements of (\dot L_2^{(m)}(\Omega,E_n)) under the mapping (\Pi); (U_2^{(m)}(\Omega)=L_2^{(m)}(E_n)\ominus \dot L_2^{(m)}(E_n\setminus\Omega,E_n)); (R) is the restriction of (\Pi) to (U_2^{(m)}(\Omega));

[
H_2^{(m)}(\Omega)=L_2^{(m)}(\Omega)\ominus \dot L_2^{(m)}(\Omega):\quad
H_2^{(m)}(\Omega,E_n)=L_2^{(m)}(E_n)\ominus \dot L_2^{(m)}(\Omega,E_n).
\tag{3}
]

Theorem 1. If (p\in L_2), and ({l^h}) is a family of functionals (1) such that, as (h\to0), (2) is fulfilled with (B=L_2(E_n)), then (2) is also fulfilled for (B=L_2^{(m)}(\Omega)).

Lemma 1. If (p), ({l^h}) satisfy the conditions of Theorem 1, then (2) is fulfilled for (B=\dot L_2^{(m)}(\Omega,E_n)).

We outline the proof of the lemma. Since
[
|l^h|{L_2^{(m)*}(E_n)}\geqslant |l^h|,}(\Omega,E_n)
]
it suffices to prove the existence of ({V_h}\subset \dot L_2^{(m)}(\Omega,E_n)) such that, as (h\to0),
[
\int_{\Omega} pV_h\,dx=|p|{L_2}\sqrt{B+o(h^m):\quad V_h(hH\gamma)=0.}(H)}\,h^m|V_h|_{L_2^{(m)}(E_n)
\tag{4}
]

For (p=1), the existence of the indicated ({V_h}) was proved in (7). It is not difficult to construct the necessary ({V_h}) if (p) is piecewise constant in (\Omega). Let now (g\in L_2) be arbitrary, (p) piecewise constant, and let ({V_h}\subset \dot L_2^{(m)}(\Omega,E_n)) satisfy (4); let ({I^h}) be a uniformly distributed family of interpolation operators (for the definition of such operators see (6)), interpolating at the points (h\gamma). Then

[
I=\int_{\Omega} gV_h\,dx
=\int_{\Omega}(g-p)(V_h-I^hV_h)\,dx+\int_{\Omega}pV_h\,dx
=I_1+I_2.
]

From Theorem 1 in (6) it follows that if (p) is close to (g) in the metric of (L_2), then (I_1) is small and (I\cong I_2). Using the approximate equality obtained and taking into account the density of piecewise constant functions in (L_2), one can prove the lemma in the general case.

We proceed directly to the proof of the theorem. From (3) it follows that

[
|l^h|{L_2^{(m)}(\Omega)}^2
=|l^h|_{\dot L_2^{(m)}(\Omega)}^2+|l^h|{H_2^{(m)}(\Omega)}^2:\quad
|l^h|_{L_2^{(m)}(E_n)}^2
]
[
=|l^h|_{\dot L_2^{(m)}(\Omega,E_n)}^2
+|l^h|_{H_2^{(m)}(\Omega,E_n)}^2.
\tag{5}
]

Lemma 1 and (5) give

[
|l^h|{L_2^{(m)}(\Omega)}^2
=B_{h,m}(H)|p|{L_2}^2h^{2m}+o(h^{2m})
+|l^h|{H_2^{(m)}(\Omega)}^2;\quad
|l^h|).}(\Omega,E_n)}^2=o(h^{2m
\tag{6}
]

If we show that

[
|l^h|_{H_2^{(m)}(\Omega)}\leqslant K|l^h|_{H_2^{(m)}(\Omega,E_n)},
\tag{7}
]

where (K) is independent of (h), then Theorem 1 will follow from (6), (7).

All elements of (L_2^{(m)}(\Omega)) can be extended to elements of (L_2^{(m)}(E_n)) (see, for example, ((^{1,8}))) and (U_2^{(m)}(\Omega)), whence, as is not difficult to see, there follows the existence for (R) of a single inverse operator (R^{-1}), defined on all of (L_2^{(m)}(\Omega)). From Banach’s theorem on inverse operators it follows that (R^{-1}) is bounded as an operator acting from (L_2^{(m)}(\Omega)) into (U_2^{(m)}(\Omega)). The boundedness of (R^{-1}) and the fact that all possible extensions of elements of (H_2^{(m)}(\Omega)) to elements of (U_2^{(m)}(\Omega)) belong to (H_2^{(m)}(\Omega,E_n)) give (7) and the assertion of Theorem 1.

Corollary. Let ({l^h}) be asymptotically optimal in (L_2^{(m)}(E_n)). Then ({l^h}) satisfies the conditions of Theorem 1 (6) (for the case (p=1), see ((^{2,5}))). From this theorem and the inequality

[
|l^h|_{L_2^{(m)*}(\Omega)} \geq |l^h|
]

it follows that ({l^h}) is asymptotically optimal in (L_2^{(m)}(\Omega)).

Remark. If (p=1) and ({l^h}) is the collection of error functionals of formulas with a regular boundary layer, then the left-hand side of (2), for (B=L_2^{(m)}(\Omega)), is (O(h^{2m+1})), as (h\to0).

The proof of this assertion is analogous to the proof of Theorem 1.

The author thanks S. L. Sobolev for his attention to the present work.

Novosibirsk
State University

Received
19 V 1969

REFERENCES

({}^{1}) S. L. Sobolev, Lectures on the Theory of Cubature Formulas, Part I, Novosibirsk, 1964.
({}^{2}) S. L. Sobolev, Lectures on the Theory of Cubature Formulas, Part II, Novosibirsk, 1965.
({}^{3}) S. L. Sobolev, DAN, 137, No. 3, 527 (1961).
({}^{4}) S. L. Sobolev, DAN, 163, No. 3, 587 (1965).
({}^{5}) S. L. Sobolev, DAN, 164, No. 2, 281 (1965).
({}^{6}) V. I. Polovinkin, DAN, 179, No. 3, 542 (1968).
({}^{7}) V. I. Polovinkin, Matem. zametki, 5, No. 3, 317 (1969).
({}^{8}) A. R. Calderon, Conf. on Partial Differential Equations, University of California, 1960.