Cubature formulas in numerical analysis and Euclidean tight designs

In combinatorics, the concept of Euclidean tt-design was first defined by Neumaier–Seidel (1988) [25], as a two-step generalization of the concept of spherical tt-design. It is possible to regard Euclidean tt-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 tt-designs. In particular, they clarify the question of what is the right definition of tight Euclidean tt-designs (tight tt-designs on RnRn and tight tt-designs on pp-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 tt-designs to the analysts. The second purpose of this paper is to review the developments of the research on tight Euclidean tt-designs. There are many new interesting examples and rich theories on tight Euclidean tt-designs. We discuss the tight Euclidean tt-designs in R2R2 carefully, and we discuss what will be the next stage of the study on tight Euclidean tt-designs. Also, we investigate the correspondence of the known examples of tight Euclidean tt-designs with the Gaussian tt-designs.