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 ${\mathbb{Z}}_2$ 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 ${\Pi }_n(V)$, 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 ${\Lambda }_{n,\gamma }^V\ge c_{\gamma }{n}^{\gamma +1}$ for every $\gamma >-1$. For nonnegative half-integer $\gamma$, the matching upper bound holds, giving the sharp universal order ${\Lambda }_{n,\gamma }^V\asymp n^{\gamma +1}$; in particular, ${\Lambda }_n^V\asymp n$ when $\gamma =0$. 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.