On almost tight Euclidean designs for rotationally symmetric integrals

Abstract

We present a characterization theorem of almost tight Euclidean $$(2e+1)$$ -designs supported by $$\lfloor e/2 \rfloor + 2$$ concentric circles in terms of roots of quasi-orthogonal polynomials. We also prove that for any $$e \ge 5$$ there exist no almost tight $$(2e + 1)$$ -designs supported by $$\lfloor e/2 \rfloor + 2$$ concentric circles for Gaussian integration. Our characterization theorem is an analogue of some previous works as such by Verlinden and Cools (Numer Math 61:395–407, 1992), Cools and Schmid (Numerical integration, IV, Oberwolfach, 1992, Birkhauser, Basel, pp 57–66, 1993) and the present authors (2010), and the nonexistence theorem provides an answer to the inverse problem for Euclidean designs, posed by Bannai et al. (Eur J Combin 31:419–422, 2010). Furthermore, the present paper includes a short review of a relationship between Euclidean designs for rotationally symmetric integrals and kernel approximation in machine learning, together with some new observations.