Weighted product Gauss quadrature after the Duffy transform

Assume Countable Choice. Let $\alpha <2$, put $\rho =1-\alpha >-1$, and let $n\ge 1$. For a continuous amplitude $G$ on $[0,1{]}^2$, let $(z_j,w_j{)}_{j=1}^n$ be the Gauss–Jacobi rule for weight $(1+t{)}^{\rho }$ on $[-1,1]$, and $({\zeta }_i,v_i{)}_{i=1}^n$ the rule for weight $1$ there. Map nodes and weights to $[0,1]$ by $r_j=(1+z_j)\mathrm{/}2,a_j=2^{-(\rho +1)}{w}_j,s_i=(1+{\zeta }_i)\mathrm{/}2,b_i=v_i\mathrm{/}2.$ The Duffy weighted product rule is $Q_{n,\alpha }^D(G)={\sum }_{j=1}^n{\sum }_{i=1}^n{a}_j{b}_{i}G(r_j,s_i)\mathrm{.}$ It approximates ${\int }_0^1{\int }_0^1{r}^{1-\alpha }G(r,s)dsdr$. When $G(r,s)=(1+s^2{)}^{-\alpha \mathrm{/}2}q(r,rs)$ for a continuous $q:T\to \mathbb{R}$, this target is the triangle integral identified in The Duffy map turns a vertex singularity into an integrable radial weight. The finite nodes and weights are well-defined by the supplied Gauss–Jacobi rule; the error estimate for such transformed amplitudes is The Duffy weighted product rule has an explicit bound using pure coordinate derivatives ↗.