Cubature formulas in numerical analysis and Euclidean tight designs

In combinatorics, the concept of Euclidean t -design was first defined by Neumaier–Seidel (1988) [25], as a two-step generalization of the concept of spherical t -design. It is possible to regard Euclidean t -design as a special case of general cubature formulas in analysis. We point out that the works on cubature formulas by Möller and others (which were not well aware by combinatorialists), are very important for the study of Euclidean t -designs. In particular, they clarify the question of what is the right definition of tight Euclidean t -designs (tight t -designs on R n and tight t -designs on p -concentric sphere). So, the first purpose of this paper is to tell combinatorialists, the importance of the theory on cubature formulas in analysis. At the same time we think that it is important for us to communicate our viewpoint of Euclidean t -designs to the analysts. The second purpose of this paper is to review the developments of the research on tight Euclidean t -designs. There are many new interesting examples and rich theories on tight Euclidean t -designs. We discuss the tight Euclidean t -designs in R 2 carefully, and we discuss what will be the next stage of the study on tight Euclidean t -designs. Also, we investigate the correspondence of the known examples of tight Euclidean t -designs with the Gaussian t -designs.