Let $f:[a,b]\to \mathbb{R}$ a smooth function. Consider a partition $a=x_1<x_2<\ldots<x_n=b$. If we put $X=(f(x_1), f(x_2), \ldots, f(x_n))$, where $x_{i+1}-x_i=\Delta x$ then:
$ (f'(x_1), f'(x_2), \ldots, f'(x_n)) \approx \frac{1}{2 \Delta x}A \cdot X $
where A is a tridiagonal matrix $(-1, 0, 1)$ (I only use finite difference there).
My questions are:
-what would be the relation between $A$ (use for approximate first derivative) and $A^{-1}$?
-Its posible that $A^{-1}$ be a way for approximate integrals via... $A^{-1} X \approx \int f$ ?
-Is there any class of finite differences that have already been studied that treat this?
Thanks.
$A$ is not square matrix because $AX = f_{n+1} - f_{n-1}$ for $n>1$ and the matrix A has n-2 rows and n columns and $A^{-1}$ don't existe you have to considerer $A^+$. For this problem is good make smalls examples for understand the structure of problem :)