Stability barriers to the construction of { ρ,σ }-reducible and fractional quadrature rules

We consider the construction of families of quadrature rules for discretising convolution integrals. Such rules arise naturally from certain linear multistep formulae for initial value problems of first and second order. We show that the choice of LMF is limited by the imposition of certain stability conditions which are appropriate when employing the formulae to discretize integral equations.