We are given an assignment problem with goes like this:
Consider a function $f:[0,1]\rightarrow \mathbb{R}$. Divide $[0,1]$ into $N$ bins: $\left[0,\frac{1}{N}\right),\left[\frac{1}{N},\frac{2}{N} \right), \ldots , \left[1-\frac{1}{N},1\right)$ and approximate $f$ by its values $\{f_n\}$ at the centre of the $n^\text{th}$ bin.
- Write down the $N\times N$ matrix such that $D$ such that $$D\begin{bmatrix} f_1 \\ \vdots\\ f_N \end{bmatrix}$$ gives an approximation to $df/dx$.
- Now go back to the continuum description. Show that as $N\rightarrow \infty$ the operator $D$ becomes an operator $\mathcal{D}$ with the property $$\int \mathcal{D}(x-x')f(x') \, dx'=\frac{df}{dx}$$
- Express $\mathcal{D}(x-x')$ in terms of the Dirac Delta function.
What I have done so far,
The derivative defined as $$\frac{df}{dx}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ In this case, $$\left.\frac{df}{dx}\right|_{i^\text{th} \text{bin}}=\lim_{N\rightarrow \infty}\frac{f_{i+1}-f_i}{1/N}=\lim_{N\rightarrow \infty}N(f_{i+1}-f_i)$$ So I can write the derivative operator as matrix like $$D=\begin{pmatrix} -1 & 1 & 0& \ddots \\ 0 & -1 & 1 & \ddots \\ \ddots&0&-1&\ddots \end{pmatrix}$$
I don't know, How to start the second part? I don't understand, How do $D$ go to $\mathcal{D}(x-x')$?
I tried going backward that is $$\int \mathcal{D}(x-x')f(x') \, dx'=\frac{df}{dx}\rightarrow \sum_i D(x-x_i) f(x_i) \, \Delta x=\frac{df}{dx}$$ I don't know what to do know? Please help me with this.