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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f\in BV(\mathbb{R}) \cap L^1(\mathbb{R}).$ Now, since $f\in BV(\mathbb{R})$ pointwise values makes and consequently we can define numerical approximations of $\int\limits_{\mathbb{R}}f(x)dx,$ via various well known formulas such as rectangle rule, midpoint rule, trapozoidal rule, simsons rule etc.

Do they still converge ($f$ need not be continuous) as in the case of sufficiently smooth $f$?

If so, what is the rate of the convergence? Is it same as what we get in the smooth case?

P.S.: I feel it works at-least for rectangle and midpoint rule because $BV$ functions satisfy $\left| \int f(x) dx - \int f(x-h)dx \right|\leq \int\limits_{\mathbb{R}}|f(x+h)-f(x)| \leq Ch.$

Thanks in advance.

Celestina
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  • do you mean by $BV$ bounded variations? what definition are you using for that? – NotaChoice Sep 28 '22 at 13:38
  • The classical one.i.e. sup of the variation of f wrt any partition is finite – Celestina Sep 30 '22 at 13:13
  • It seems that you are confusing terminology still, $L^1$ is the class of all 'integrable' functions, and hence include non continuous ones, meaning the integral here converges. The rate of convergence of a numerical scheme will depend on the scheme and how the algorithm is set up, at first glance I don't think that the bounded variations will intervene here, unless maybe if you push the analysis deeper for certain schemes. – NotaChoice Oct 03 '22 at 22:46
  • Okay..let me put it this way. Is there any class of functions (bigger than the set of continuous functions) for which the convergence of these formulas can be proved. – Celestina Oct 04 '22 at 09:28
  • All $L^1$ functions can be subject to appropriate numerical schemes to compute the integral. – NotaChoice Oct 05 '22 at 13:42
  • Can you please share the reference, where this has been proved explicitly for L1 functions(not necessarily differentiable)? – Celestina Oct 10 '22 at 06:44

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