Summation Expression with Range and Sequence expression

Question

I've been trying to search for the meaning for this summation for a couple of days now. I've used https://approach0.xyz/ to search on the stack, but I cannot seem to find an explanation that helps me.

All I know that it is supposed to be a spline equation, but I don't understand, with my limited math knowledge, the range and sequence sign in this summation.

$g(u)=\sum_{i=0}^{\ n\ }\ 1\left\{\left[u_i,u_{i+1}\right)\right\}s_i\left(u\right)$

$s_i(u)= d_i(u - u_i)^3 + c_i(u-u_i)^2 + b_i(u-u_i)+a_i$

Full Equation Here from paper

thank you

Answer 1

Given a set $X$, the indicator function or characteristic function of a set $A\subseteq X$ is \begin{align*} \mathbb{1}_{A}= \begin{cases} 1&\qquad x\in A\\ 0&\qquad x\not \in A \end{cases} \end{align*} Here we consider the set $X=\mathbb{R}$ of real numbers and for each interval $\left[u_i,u_{i+1}\right)\subseteq \mathbb{R}$, $0\leq i\leq n$ we consequently have \begin{align*} \mathbb{1}_{\left[u_i,u_{i+1}\right)}= \begin{cases} 1&\qquad x\in \left[u_i,u_{i+1}\right)\\ 0&\qquad x\not \in \left[u_i,u_{i+1}\right) \end{cases}\tag{1} \end{align*}

The function $g$ is given in terms of (1). It can be written as \begin{align*} g(u)&=\sum_{i=0}^n\mathbb{1}_{\left[u_i,u_{i+1}\right)}s_i(u)\\ &=\mathbb{1}_{\left[u_0,u_{1}\right)}s_0(u)+\mathbb{1}_{\left[u_1,u_{2}\right)}s_1(u) +\cdots+\mathbb{1}_{\left[u_n,u_{n+1}\right)}s_n(u)\\ &=\begin{cases} s_0(u)&\qquad x\in\left[u_0,u_{1}\right)\\ s_1(u)&\qquad x\in\left[u_1,u_{2}\right)\\ \quad\vdots&\\ s_n(u)&\qquad x\in\left[u_n,u_{n+1}\right)\\ \end{cases} \end{align*}

Note, the usage of braces $1_{\{[u_i,u_{i+1}\}}$ surrounding the interval is not correct, since each of the intervals $[u_i,u_{i+1})$ is already the set we want to use.