Asymptotically short generalizations of t-design curves

2025/05/05 by Lindblad, Ayodeji
#05B30 #41A55 (primary) #41A63 #41A81 #65D30 #65D32 (secondary) #FOS: Mathematics #Metric Geometry (math.MG) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2505.03056

Abstract

Ehler and Gröchenig posed the question of finding t-design curves γ_t\unicodex2013curves whose associated line integrals exactly average all degree at most t polynomials\unicodex2013on S^d of asymptotically optimal arc length ℓ(γ_t)\asymp t^d-1 as t→∞. This work investigates analogues of this question for weighted and ε_t-approximate t-design curves, proving existence of such curves γ_t on S^d of arc length ℓ(γ_t)\asymp t^d-1 as t→∞ for all d∈\Bbb N₊ in the weighted setting (in which case such curves are asymptotically optimal) and all odd d∈\Bbb N₊ in the approximate setting (where we have ε_t\asymp1/t as t→∞). Formulas for such weighted t-design curves for d∈\2,3\ are presented.

Citations

• Fixed-strength spherical designs
• Geodesic cycles on the Sphere: t-designs and Marcinkiewicz-Zygmund Inequalities
• Asymptotically optimal t-design curves on S³
• Designs related through projective and Hopf maps
• t-design curves and mobile sampling on the sphere
• Efficient Spherical Designs with Good Geometric Properties
• Relation between spherical designs through a Hopf map
• Spherical t_ε-Designs for Approximations on the Sphere
• Optimal asymptotic bounds for spherical designs
• Numerical cubature from Archimedes' hat-box theorem
• McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions

Related

• Asymptotically optimal t-design curves on S³
• Short spherical t-design curves
• t-design curves and mobile sampling on the sphere
• Riemann hypothesis and the arc length of the Riemann Z(t)-curve
• L^p-improving estimates for averages on polynomial curves