A recursive construction of projective cubature formulas and related isometric embeddings

TL;DR

This paper introduces a recursive method to construct projective cubature formulas on spheres over real, complex, or quaternionic fields, leading to new bounds for minimal node counts and related isometric embeddings.

Contribution

It provides a novel recursive construction technique for projective cubature formulas, improving bounds and understanding of isometric embeddings in various field settings.

Findings

New upper bounds for minimal nodes in cubature formulas

Enhanced understanding of isometric embeddings between normed spaces

Recursive construction method applicable to real, complex, and quaternionic spheres

Abstract

A recursive construction is presented for the projective cubature formulas of index $p$ on the unit spheres $S(m,K)\subset K^m$ where $K$ is $R$ or $C$, or $H$. This yields a lot of new upper bounds for the minimal number of nodes $n=N_K(m,p)$ in such formulas or, equivalently, for the minimal $n$ such that there exists an isometric embedding $\ell_{2; K}^m \rightarrow \ell_{p; K}^n$.