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Bla Bajnok

On Euclidean designs

Keywords: Euclidean design, spherical design, interval design, GaussJacobi quadrature, Gegenbauer polynomial, tight design, harmonic polynomial

A Euclidean t-design, as introduced by Neumaier and Seidel (1988), is a finite set 𝒳 ⊂ ℝⁿ with a weight function w : 𝒳 → ℝ⁺ for which holds for every polynomial ƒ of total degree at most t; here R is the set of norms of the points in 𝒳, W_r is the total weight of all elements of 𝒳 with norm r, S_r is the n-dimensional sphere of radius r centered at the origin, and is the average of ƒ over S_r. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), also proved a Fisher-type inequality |𝒳| ≥ N(n, |R|, t) (assuming that the design is antipodal if t is odd). For fixed n and |R| we have N(n, |R|, t) = O(tⁿ⁻¹).

This paper contains two main results. First, we provide a recursive construction for Euclidean t-designs in ℝⁿ. Namely, we show how to use certain Gauss–Jacobi quadrature formulae to ‘lift’ a Euclidean t-design in ℝⁿ⁻¹ to a Euclidean t-design in ℝⁿ, preserving both the norm spectrum R and the weight sum W_r for each r ∈ R. For fixed n and |R| this construction yields a design of size O(tⁿ⁻¹); however, the coefficient of tⁿ⁻¹ here is significantly greater than it is in N(n, |R|, t).

A Euclidean design with exactly N(n, |R|, t) points is called tight; in both of the above mentioned papers it was conjectured that a tight Euclidean design with t ≥ 4 must be a spherical design, that is, |R| = 1 and w is constant on 𝒳. Bannai and Bannai (2003) disproved this conjecture by exhibiting an example for the parameters, (n, |R|, t) = (2,2,4). Here we construct tight Euclidean designs for n = 2 and every t and |R| with |R| ≤ .

Advances in Geometry, Walter de Gruyter

Print ISSN: 1615-715X
Volume: 6, 07/2006
Pages: 423 - 438

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