Convergence properties of a class of product formulas for weakly singular integral equations

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type

$$\int_{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1,$$

where the kernel $K(x,y)$ is weakly singular and the function $f(x)$ has singularities only at the endpoints $\pm 1$. In particular, when $K(x,y) = \log \vert x - y\vert$, $K(x,y) = \vert x - y{\vert^v}$, $v > - 1$, and $f(x)$ has algebraic singularities of the form ${(1 \pm x)^\sigma }$, $\sigma > - 1$, we prove that the uniform rate of convergence of the rules is $O({m^{ - 2 - 2\sigma }}{\log ^2}m)$ in the case of the first kernel, and $O({m^{ - 2 - 2\sigma - 2v}}\log m)$ if $v \leq 0$, or $O({m^{ - 2 - 2\sigma }}\log m)$ if $v > 0$, for the second, where m is the number of points in the quadrature rule.