Saturday, 18 June 2011

real analysis - Dependence of error on mesh for Riemann sums

If f(x)=x on [0,1], then a mesh less than delta means the Riemann sum differs from the integral by about delta/2, so delta(epsilon) is about 2epsilon. Thus, I presume that you mean epsilon instead of epsilon2.



A function with bounded variation satisfies a linear upper bound for the error in terms of the mesh size and total variation, hence a linear lower bound for the required delta(epsilon).



A function without bounded variation may require a smaller mesh. For example, consider the continuous extension of
f(x)=xcos(pi/xn) to [0,1]. The difference in f between sampling f at x=(2k+1)1/n and at
x=(2k)1/n is about 2x at locations differing by about 1/n xn+1. That means with mesh delta, you can sample f to be either 1 or 1 up to about x=cdelta1/(n+1), for an error of cdelta2/(n+1).



Therefore, delta(epsilon) can't be larger than cepsilon(n+1)/2.

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