Non-uniform discretization

Question

Let $x \in [0, R]$. I want to discretize this variable in a way s.t. each interval is half wide than the previous one. How can I give an expression of $x_i$? All I'm given is that

$\displaystyle \sum_{i=0}^{N-1} \dfrac{1}{2^i} = 2^{1-N} (-1 + 2^N)$.

Answer 1

Suppose you have $N$ intervals in total, and they are numbered as $I_i$, $i = 0, \ldots, N - 1$, in an increasing order. The width of interval $I_0$ is defined as $L_0$, then width of $I_i$ is $L_i = L_0/2^i$. So we have $$ \sum_{i = 0}^{N - 1}{L_i} = L_0\sum_{i = 0}^{N - 1}{\frac{1}{2^i}} = R. $$ With the equation your are given, you can get $L_0 = 2^{N - 1}R/(2^N - 1)$, and thus $L_i, i = 1, \ldots, N - 1$, which completes the discretization.