Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface
Abstract
We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is formulated as a measure-preserving ℤ2 quotient. It identifies degree-m conic traces with even disk polynomials of degree at most 2m and gives a two-way correspondence between positive degree-2n cone cubature and centrally symmetric positive degree-(4n+1) disk cubature. Consequently, Möller's lower bound and the node excess above it transfer exactly, producing compressed non-product cone rules from near-minimal disk formulas. The quotient also transfers reproducing kernels and hyperinterpolation operators. Every positive degree-2n cone rule yields an exact L2 sampling isometry on Πn(V), so the weighted sampling matrix has condition number one, independently of the number and geometry of its nodes. The associated Lebesgue constants satisfy the rule-independent bound Λn,γV≥ cγnγ+1 for every γ>−1. For nonnegative half-integer γ, the matching upper bound holds, giving the sharp universal order Λn,γVasymp nγ+1; in particular, ΛnVasymp n when γ=0. Thus node compression preserves exact L2 conditioning and, in this parameter range, optimal-order L∞ stability. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.
Related
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BibTeX
@misc{an2026positive,
title = {{Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface}},
author = {Congpei An and Yan Ge},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36430},
eprint = {2609.36430},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36430},
}
RIS
TY - UNPB
TI - Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface
AU - Congpei An
AU - Yan Ge
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.36430
UR - https://arxiv.org/abs/2609.36430
AB - We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is formulated as a measure-preserving ℤ_2 quotient. It identifies degree-m conic traces with even disk polynomials of degree at most 2m and gives a two-way correspondence between positive degree-2n cone cubature and centrally symmetric positive degree-(4n+1) disk cubature. Consequently, Möller's lower bound and the node excess above it transfer exactly, producing compressed non-product cone rules from near-minimal disk formulas. The quotient also transfers reproducing kernels and hyperinterpolation operators. Every positive degree-2n cone rule yields an exact L^2 sampling isometry on Π_n(V), so the weighted sampling matrix has condition number one, independently of the number and geometry of its nodes. The associated Lebesgue constants satisfy the rule-independent bound Λ_n,γ^V≥ c_γn^γ+1 for every γ>−1. For nonnegative half-integer γ, the matching upper bound holds, giving the sharp universal order Λ_n,γ^Vasymp n^γ+1; in particular, Λ_n^Vasymp n when γ=0. Thus node compression preserves exact L^2 conditioning and, in this parameter range, optimal-order L^∞ stability. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.
ER -