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A lower bound for the number of nodes in the cubature formula for a centrally symmetric integral. (English. Russian original) Zbl 0721.41042

Math. Notes 45, No. 5, 396-400 (1989); translation from Mat. Zametki 45, No. 5, 70-75 (1989).
By an integral \({\mathcal I}_ M\) on an affine algebraic manifold \(M\subseteq {\mathbb{R}}^ n\) (M\(\neq \emptyset)\) we mean a linear functional \({\mathcal I}_ M: {\mathbb{R}}[x]\to {\mathbb{R}}\), possessing the additional property: I) \({\mathcal I}_{M| P^+(M)}\geq 0\), and, moreover, for Ker \({\mathcal I}_ M\cap P^+(M)={\mathcal A}_ M\). Here \(x=(x_ 1,...,x_ n)\), \(P^+(M)=\{f\in {\mathbb{R}}[x]:\) \(f|_ M\geq 0\}\) \({\mathcal A}_ M=\{f\in {\mathbb{R}}[x]:\) \(f|_ M\equiv 0\}\) is an ideal in the ring \({\mathbb{R}}[x]\), corresponding to the manifold M (such ideals are said to be real). We consider the cubature formulas \((1)\quad {\mathcal I}_ M=c_ 1{\mathcal I}_{\xi^{(1)}}+...+c_ N{\mathcal I}_{\xi^{(N)}}+r,\) where \(c_ 1,...,c_ N\in {\mathbb{R}}\); \(\xi^{(1)},...,\xi^{(N)}\) are pairwise distinct points in M, \({\mathcal I}_{\xi^{(1)}},...,{\mathcal I}_{\xi^{(N)}}\) are linear functionals of the form \({\mathcal I}_{\xi}: f\in {\mathbb{R}}[x]\to f(\xi).\)
Let \(P^ t_ n\subset {\mathbb{R}}[x]\) be the linear subspace of all polynomials of degree at most t \((t\in N_ 0)\); \(H_ t=H(t;{\mathcal A}_ M)\) is the Hilbert function of the nonhomogeneous ideal \({\mathcal A}_ M\), which can be defined by means of the equalities \[ H(t;{\mathcal A}_ M)=\dim P^ t_ n/{\mathcal A}_ M\cap P^ t_ n,\quad t\in N_ 0;\quad =0,\quad t\in -N. \] We say that formula (1) possesses the m-property if \(r|_{P^ m_ n}\equiv 0\). The author [VINITI, No.5272-B87 (1987)], for the number N of nodes in the cubature formula (1), possessing the m-property, has established the “simplest” bound (2) \(N\geq H([m/2];{\mathcal A}_ M).\)
We discuss one such bound, obtained by H. Möller [Numerische Integration, Tag. Obewolfach 1978, ISNM Vol. 45, 221-230 (1979; Zbl 0416.65019)] for the case when m is an odd number and the integral \({\mathcal I}_ M\) is “centrally symmetric”: \({\mathcal I}_ M(f(x))={\mathcal I}_ M(f(-x))\) for all \(f\in {\mathbb{R}}[x]\).

MSC:

41A55 Approximate quadratures

Citations:

Zbl 0416.65019
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References:

[1] G. G. Rasputin, ?Some questions in the algebraic theory of cubature formulas,? Manuscript desposited at VINITI, No. 5272-B 87 (1987).
[2] B. Renschuch, Elementare und Praktische Idealtheorie, VEB Deutscher Verlag der Wissenschaften, Berlin (1976). · Zbl 0354.13001
[3] I. P. Mysovskikh, Interpolational Cubature Formulas [in Russian], Nauka, Moscow (1981). · Zbl 0537.65019
[4] H. M. Möller, ?Lower bounds for the number of nodes in cubature formulae,? in: Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978), Internat. Ser. Numer. Math., No. 45, Birkhauser, Basel (1979), pp. 221-230.
[5] J.-J. Risler, ?Une caractérisation des idéaux des variétés algébriques réeles,? C. R. Acad. Sci. Paris, Ser. A-B,271, No. 23, A1171-A1173 (1970). · Zbl 0211.53401
[6] H. M. Möller, ?Kubaturformeln mit minimaler Knotenzahl,? Numer. Math.,25, No. 2, 185-200 (1976). · Zbl 0319.65019 · doi:10.1007/BF01462272
[7] F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Univ. Press (1916). · JFM 46.0167.01
[8] I. P. Mysovskikh, ?On the calculation of integrals over the surface of the sphere,? Dokl. Akad. Nauk SSSR,235, No. 2, 269-272 (1977). · Zbl 0377.65011
[9] M. V. Noskov, ?Cubature formulas for the approximate integration of periodic functions,? in: Metody Vychisl., No. 14, Leningrad State Univ. (1985), pp. 15-23. · Zbl 0754.41030
[10] I. P. Mysovskikh, ?On cubature formulas that are exact for trigonometric polynomials,? Dokl. Akad. Nauk SSSR,296, No. 1, 28-31 (1987).
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