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Real and complex spherical designs and their Gramian

Shayne Waldron

TL;DR

This work develops a Gramian-based variational framework to characterise real and complex spherical designs for unitarily invariant polynomial spaces, unifying their treatment via a potential $A_{w,c}$ built from reproducing kernels. When $K(x,y)$ depends only on $\langle x,y\rangle$, the potential reduces to a Gramian function whose zeros identify designs, enabling both analytic characterisations (e.g., for $t$-designs, half-designs, and projective designs) and numerical construction through minimisation. The paper systematically derives real and complex design theories, including Gegenbauer polynomial orthogonality, product rules, and canonical potentials for various design classes, and develops upper and lower bounds on the number of design points (DGS bounds) with extensions to complex and projective settings. It also introduces and exploits operator-rank and annihilator techniques to obtain absolute and special bounds for complex designs, including two- and three-angle configurations, and connects these to classical polynomials such as Jacobi and Gegenbauer, yielding new insights and tight examples. Overall, the framework provides a unified, Gramian-driven approach that advances both the theory and construction of real, complex, and projective spherical designs with practical implications for cubature rules and frame theory.

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.

Real and complex spherical designs and their Gramian

TL;DR

This work develops a Gramian-based variational framework to characterise real and complex spherical designs for unitarily invariant polynomial spaces, unifying their treatment via a potential built from reproducing kernels. When depends only on , the potential reduces to a Gramian function whose zeros identify designs, enabling both analytic characterisations (e.g., for -designs, half-designs, and projective designs) and numerical construction through minimisation. The paper systematically derives real and complex design theories, including Gegenbauer polynomial orthogonality, product rules, and canonical potentials for various design classes, and develops upper and lower bounds on the number of design points (DGS bounds) with extensions to complex and projective settings. It also introduces and exploits operator-rank and annihilator techniques to obtain absolute and special bounds for complex designs, including two- and three-angle configurations, and connects these to classical polynomials such as Jacobi and Gegenbauer, yielding new insights and tight examples. Overall, the framework provides a unified, Gramian-driven approach that advances both the theory and construction of real, complex, and projective spherical designs with practical implications for cubature rules and frame theory.

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.

Paper Structure

This paper contains 10 sections, 26 theorems, 250 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. The variational characterisation of designs
  4. Real spherical designs
  5. Complex spherical designs
  6. Projective spherical designs
  7. Orthogonality of the Gegenbauer polynomials
  8. Products of the Gegenbauer polynomials
  9. Bounds for real and complex spherical designs
  10. Absolute and special bounds for complex designs

Key Result

Theorem 3.1

(Variational characterisation) Let $\mu$ be a measure on $\Omega$ with $\mu(\Omega)=1$, $P$ be a finite dimensional space of integrable functions on $\Omega$, and $H$ be the subspace of functions which are orthogonal to the constants, i.e., Let $\Phi=(v_1,\ldots,v_n)$, $v_j\in\Omega$, and $w=(w_j)\in\mathbb{R}^n$ be weights with $w_1+\cdots+w_n=1$. Write $H$ as a direct sum $H=\oplus_\ell H^{(\el

Theorems & Definitions (50)

  • Example 2.1
  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Lemma 4.1
  • Theorem 4.1
  • Corollary 4.1
  • Theorem 4.2
  • Example 4.4
  • ...and 40 more