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In this paper we study the existence of 2-dimensional minimal cubature formulae of degree 2e + 1 for a circularly symmetric integral that are supported by e/2 + 1 concentric spheres. We give powerful necessary conditions for the existence of such formulae by blending the theory of Euclidean designs and the theory of orthogonal polynomials. And thereby we show that there exists no minimal formula of degree 2e + 1 for the Gaussian integral supported by e/2 + 1 concentric spheres for any e ≥ 3.

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