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 used in a basis-free form as a measure-preserving $\mathbb Z_2$ quotient. It identifies the full degree-$m$ conic trace space with the even disk polynomials of degree at most $2m$ and converts positive degree-$2n$ cone cubature into centrally symmetric positive degree-$(4n+1)$ disk cubature, and conversely. This yields an exact transfer of Möller's lower bound and of the node excess above it, so that near-minimal disk formulas produce compressed non-product cone rules. The same quotient transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule gives an exact $L^2$ sampling isometry on $Π_n(V)$; in particular, the weighted sampling matrix has condition number one, independently of the number and geometry of the nodes. For the unweighted radial case $γ=0$, if $Λ_n^V$ denotes the $C(V)\to C(V)$ Lebesgue constant of degree-$n$ hyperinterpolation, then every positive degree-$2n$ cone cubature rule satisfies the rule-independent sharp law $$c n \le Λ_n^V \le C n.$$ More generally, the upper bound $Λ_{n,γ}^V \le C_γn^{γ+1}$ holds when $γ$ is a nonnegative half-integer. Thus node compression preserves exact $L^2$ conditioning while the associated $L^\infty$ stability has the optimal universal order. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.