Abstract
In this paper we prove that there exists no minimum cubature formula of degree 4k and 4k+2 for Gaussian measure on ℝ2 supported by k+1 circles for any positive integer k, except for two formulas of degree 4.
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Bannai, Ei., Bannai, Et.: Tight Gaussian 4-designs. J. Algebr. Comb. 22, 39–63 (2005)
Bannai, Ei., Bannai, Et.: On Euclidean tight 4-designs. J. Math. Soc. Jpn. 58, 775–804 (2006)
Bannai, Ei., Bannai, Et.: A survey on spherical designs and algebraic combinatorics on spheres. Eur. J. Comb. 30, 1392–1425 (2009)
Bannai, Ei., Bannai, Et., Hirao, M., Sawa, M.: Cubature formulas in numerical analysis and Euclidean tight designs. Eur. J. Comb. 31, 423–441 (2010) (Special Issue in honour of Prof. Michel Deza)
Bannai, Et.: On antipodal Euclidean tight (2e+1)-designs. J. Algebr. Comb. 24, 391–414 (2006)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol. 13. Gordon and Breach, New York (1978)
Cools, R., Schmid, H.J.: A new lower bound for the number of nodes in cubature formulae of degree 4n+1 for some circularly symmetric integrals. Int. Ser. Numer. Math. 112, 57–66 (1993)
Hirao, M., Sawa, M.: On minimal cubature formulae of odd degrees for circularly symmetric integrals. Adv. Geom. (to appear)
Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25, 185–200 (1975/76)
Möller, H.M.: Lower bounds for the number of nodes in cubature formulae. In: Numerische Integration, Tagung, Math. Forschungsinst., Oberwolfach, 1978. Internat. Ser. Numer. Math., vol. 45, pp. 221–230. Birkhäuser, Basel (1979)
Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs (1971)
Szegő, G.: Orthogonal Polynomials. AMS Colloquium Publications, vol. 23 (1939)
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Bannai, E., Bannai, E., Hirao, M. et al. On the existence of minimum cubature formulas for Gaussian measure on ℝ2 of degree t supported by \([\frac{t}{4}]+1\) circles. J Algebr Comb 35, 109–119 (2012). https://doi.org/10.1007/s10801-011-0295-3
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DOI: https://doi.org/10.1007/s10801-011-0295-3