# Convergence of Gauss quadrature for a discontinuous function

Is there a known error estimate for Gaussian quadrature when applied to a discontinuous function?

For simple one-dimensional experiments, the error appears to be bounded by $C h$, where $C$ is some constant and $h = \max |x_{i+1}-x_i|$ is the maximum spacing between two nodes, but I haven't found a clean way to show this.

• Since Gaussian quadrature is equivalent to interpolating by 2n+1 order polynomial and then exactly integrating the result, this boils down to how Gibbs phenomenon for polynomial interpolation can affect that integral as we increase the order. If we know the Gibbs phenomenon for polynomial interpolation has bounded over/undershoot, like it does for Fourier series, then the localized area of the Gibbs phenomenon will have an integral that decays to 0 as you increase the number of interpolation points, just by virtue of the grid spacing being closer and closer together. – Reid.Atcheson Feb 8 '18 at 18:04
• i think @Reid.Atcheson hit the nail on the head with his comment. if you're looking for more precise error estimates, this paper should be a good start. – GoHokies Feb 8 '18 at 20:24
• Thanks @Reid.Atcheson. Agreed on the intuition, though I don't know how yet to massage that intuition into a concrete O(h) estimate yet. – Jesse Chan Feb 9 '18 at 18:14
• @GoHokies that's the sort of paper I'm looking for, but they consider only a > 0, which excludes the fully discontinuous case. – Jesse Chan Feb 9 '18 at 18:30