How to approximate L^1[0,1] functions?

Question

Do functions on a uniform grid with n points in the interval $[0,1]$ approximate $L^1[0,1]$ functions, as $n \to \infty$?

I want to sample functions in $L^1[0,1]$ space numerically and I want to be sure that as I make the grid finer, the space of sampled functions approach the desired space.

Answer 1

It's a bit unclear what kind of functions you meant: two interpretations are

• piecewise constant functions, where you assign a value to each of $n$ subintervals of length $1/n$.
• piecewise linear functions, where you assign a value to each of $n$ points $0, \frac{1}{n-1}, \frac{2}{n-1}, \dots, 1$ and interpolate by lines in between.

Either way, the functions you obtain indeed form a dense subset of $L^1$, meaning that for every $L^1$ function $f$ and every $\epsilon>0$ there is a function $g$ of the above form such that $\|f-g\|_{L^1}$.

The key point is that continuous functions are dense in $L^1$, and sampling a continuous function in either way will produce a sequence converging to it uniformly, hence converging in $L^1$ norm.