Year: 2026
Type: preprint
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 $\mathbb Z_2$ 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 $Λ_{n,γ}^V\ge c_γn^{γ+1}$ for every $γ>-1$. For nonnegative half-integer $γ$, the matching upper bound holds, giving the sharp universal order $Λ_{n,γ}^V\asymp n^{γ+1}$; in particular, $Λ_n^V\asymp n$ when $γ=0$. Thus node compression preserves exact $L^2$ conditioning and, in this parameter range, optimal-order $L^\infty$ stability. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.
Source: arXiv (Cornell University)
Authors: An, Congpei, Ge, Yan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214897688