Transactions of the American Mathematical Society Series B· 2026Q1
From point sets to curves: 𝑡-Designs and Marcinkiewicz-Zygmund inequalities on the sphere
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- Q1SCImago
- 2026year
Short summary
New geodesic cycles on the sphere are constructed that act as t-designs, enabling equal-weight quadrature, and others are proven to exist that satisfy Marcinkiewicz-Zygmund inequalities with near-optimal length.
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Key points
- Explicit geodesic cycles are constructed that achieve t-design properties for small t.
- The existence of geodesic cycles satisfying Marcinkiewicz-Zygmund inequalities with asymptotically optimal length is proven.
- These findings connect geometric curve constructions on the sphere to quadrature rules and approximation theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
A geodesic cycle is a closed curve that connects finitely many points along geodesics. We study geodesic cycles on the sphere in regard to their role in equal-weight quadrature via t t -designs and in approximation theory through Marcinkiewicz-Zygmund inequalities. In the first part we analyze and construct explicit geodesic cycles that lead to t t -design curves for small t t . In the second part we prove the existence of geodesic cycles satisfying Marcinkiewicz-Zygmund inequalities with asymptotically optimal length.
The authors' abstract, as published at the source. Transactions of the American Mathematical Society Series B, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics