On the evaluation of certain two-dimensional singular integrals with Cauchy kernels
The authors consider product formulas of interpolatory type for the two- dimensional Cauchy principal value integral: \[ I(f;\eta_ 1,\eta_ 2)=\int^{1}_{-1}\int^{1}_{-1}\omega(x,y)\frac{f(x,y)}{(x-\eta_ 1)(y-\eta_ 2)}dx dy \] where: \(\eta_ 1,\eta_ 2\in(-1,1)\), \(\omega(x,y)=\omega_ 1(x).\omega_ 2(y)\) and \(\omega_ 1(x)\) and \(\omega_ 2(y)\) are two absolutely integrable weight functions. The integral is approximated by \[ F_{n,m}(f;\eta_ 1,\eta_ 2)=\sum^{n}_{i=0}\sum^{m}_{j=0}A_ i^{(1)}(\eta_ 1)A_ j^{(2)}(\eta_ 2)f(x_ i,y_ j) \] where the nodes \(\{x_ i\}\) and \(\{y_ j\}\) are the zeros of the Chebyshev polynomials of the first kind, commonly named ''classical'' abscissas, or the Clenshaw points, often called ''practical'' abscissas. Convergence results for these rules are presented.
scientific article; zbMATH DE number 3859692