Real and complex spherical designs and their Gramian
Shayne Waldron
Abstract:
If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such spherical designs are determined by their Gramian (Gram matrix). We outline a general method to obtain such a characterisation as the minima of a function of the Gramian, which we call a potential. This characterisation can be used for the numerical and analytic construction of spherical designs. When the space P of polynomials is not irreducible under the action of the unitary group, then the potential is not unique. In several cases of interest, e.g., spherical t-designs and half-designs, we use this flexibility to provide potentials with a very simple form. We then use our results to develop certain aspects of the theory of real and complex spherical designs for unitarily invariant polynomial spaces.
Keywords:
Gramian (Gram matrix),
spherical t-designs,
spherical half-designs,
tight spherical designs,
finite tight frames,
integration rules, cubature rules,
cubature rules for the sphere,
reproducing kernel,
positive definite function,
Gegenbauer polynomials,
Zernike polynomials,
complex spherical design,
potential,
frame force,
codes,
Math Review Classification:
Primary 05B25, 05B30, 42C15, 65D30;
Secondary 94A12. 51M20.
Length: 45 Pages
Last Updated: 8 November 2025
Availability: