Marcinkiewicz-Zygmund inequalities for hemispherical t-designs

Abstract

In this paper, we introduce a new set of 𝕃₂-orthogonal polynomials on the hemisphere 𝕊^d₊:={x∈ℝ^d+1:|x|=1, x·e_d+1≥0} and establish 𝕃₁ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical t-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree t. We propose variational characterizations of hemispherical t-designs by the 𝕃₂-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the 𝕃₂-orthogonal polynomials, we prove that for each n≥ C_d t^d, there exists a hemispherical t-design on the hemisphere with n points, where C_d is a constant depending only on d.

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BibTeX RIS

BibTeX

@misc{li2026marcinkiewiczzygmund,
  title = {{Marcinkiewicz-Zygmund inequalities for hemispherical $t$-designs}},
  author = {Chao Li and Xiaojun Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39171},
  eprint = {2609.39171},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39171},
}

RIS

TY  - UNPB
TI  - Marcinkiewicz-Zygmund inequalities for hemispherical t-designs
AU  - Chao Li
AU  - Xiaojun Chen
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39171
UR  - https://arxiv.org/abs/2609.39171
AB  - In this paper, we introduce a new set of 𝕃_2-orthogonal polynomials on the hemisphere 𝕊^d_+:={x∈ℝ^d+1:|x|=1, x·e_d+1≥0} and establish 𝕃_1 Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical t-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree t. We propose variational characterizations of hemispherical t-designs by the 𝕃_2-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the 𝕃_2-orthogonal polynomials, we prove that for each n≥ C_d t^d, there exists a hemispherical t-design on the hemisphere with n points, where C_d is a constant depending only on d.
ER  -

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