Weighted sharp Lebesgue law for product hyperinterpolation

Establish the conjectured asymptotic order of the Lebesgue constant for the Gauss–Jacobi/trapezoidal product hyperinterpolation rule on the truncated conic surface for all weights with gamma>-1, namely \Lambda_{n,\gamma}^{V,\prod}\asymp n^{1/2} for -1<\gamma<-1/2 and \Lambda_{n,\gamma}^{V,\prod}\asymp n^{\gamma+1} for \gamma\ge-1/2.

Background

For the truncated conic surface equipped with the Jacobi-type weight parameterized by gamma>-1, the paper studies the Lebesgue constant of degree-n hyperinterpolation generated by positive degree-2n cubature rules. The authors transfer estimates from weighted disk hyperinterpolation through the quadratic disk-to-cone quotient.

The paper proves the upper bound \Lambda_{n,\gamma}^V\le C_\gamma n^{\gamma+1} for nonnegative half-integer gamma and proves the matching linear upper and lower bounds when gamma=0. It does not establish the conjectured two-regime asymptotic law for the remaining weights, which requires a weighted boundary-kernel analysis or an equivalent estimate for the even restriction of disk hyperinterpolation.