Convergence of quadrature for non-smooth integrands

Question

Standard convergence estimates for quadrature formulas $(w_i,x_x)_{i = 1}^n$ are of the form $$ \left|\int_a^b f(x) \, dx - \sum_{i = 1}^n w_i \, f(x_i)\right| \leq C (b - a)^{p+1} \max_{x \in (a,b)} |f^{(p)}(x)|, $$ see e.g.

• https://en.wikipedia.org/wiki/Trapezoidal_rule#Error_analysis,
• https://en.wikipedia.org/wiki/Simpsons_rule#Error.

What if the function is not $p$ times continuously differentiable, but only $p'<p$? Can I then assume $$ \left|\int_a^b f(x) \, dx - \sum_{i = 1}^n w_i \, f(x_i)\right| \leq C (b - a)^{p'+1} \max_{x \in (a,b)} |f^{(p')}(x)|, $$ or is there any other theory available?

Answer 1

There is good theory available for this case, too. A good paper to start with is

http://www.maa.org/sites/default/files/An_Elementary_Proof30705.pdf

You may also be interested in doing an asymptotic error analysis of your integrals, in order to obtain reliable and accurate error estimates as a priori error bounds can be very pessimistic. One reliable technique is Richardson extrapolation.