On the existence of minimum cubature formulas for Gaussian measure on \Bbb R 2 of degree t supported by [\frac t4]+1 [\frac{t}{4}]+1 circles
Eiichi Bannai
, Etsuko Bannai
, Masatake Hirao
and Masanori Sawa
DOI: 10.1007/s10801-011-0295-3
Abstract
In this paper we prove that there exists no minimum cubature formula of degree 4 k and 4 k+2 for Gaussian measure on \Bbb R 2 supported by k+1 circles for any positive integer k, except for two formulas of degree 4.
Pages: 109–119
Keywords: keywords cubature formula; Euclidean design; Gaussian design; Laguerre polynomial
Full Text: PDF
References
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2. Bannai, Ei., Bannai, Et.: On Euclidean tight 4-designs. J. Math. Soc. Jpn. 58, 775-804 (2006)
3. Bannai, Ei., Bannai, Et.: A survey on spherical designs and algebraic combinatorics on spheres. Eur. J. Comb. 30, 1392-1425 (2009)
4. 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)
5. Bannai, Et.: On antipodal Euclidean tight (2e + 1)-designs. J. Algebr. Comb. 24, 391-414 (2006)
6. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol.
13. Gordon and Breach, New York (1978)
7. 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)
8. Hirao, M., Sawa, M.: On minimal cubature formulae of odd degrees for circularly symmetric integrals. Adv. Geom. (to appear)
9. Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25, 185-200 (1975/76)
10. 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.
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