# Cubic B-Spline (Basic Spline) Non-Zero Domain

Why does the cubic B-spline (Basic Spline) takes on the value zero outside the interval $[x_i, x_{i+4}]$? Specifically, why is the lower value of the interval $x_i$?

The cubic B-spline is defined as:

$$B_i(x) = (x_{i+4} - x_i) f_x [x_i, x_{i+1}, x_{i+2}, x_{i+3}, x_{i+4}]$$

$f_x[x_i, x_{i+1}, x_{i+2}, x_{i+3}, x_{i+4}]$ is the 4-th order divided difference of $f_x(t) = (t-x)_+^3$. Note that if $x<x_i$, then still $f_x[x_i, x_{i+1}]$ will be non-zero.

Another formula for the cubic B-spline is

$$B_i(x) = (x_{i+4} - x_i) \sum_{j=i}^{j=i+4} \frac{(x_j-x)_+^3}{\psi'_i(x_j)}$$

where $\psi'_n(x_i) = (x_i - x_0) (x_i - x_1) \ldots (x_i-x_{i-1}) (x_i - x_{i+1}) \ldots (x-x_n)$

Note that if I use the second formula, I can see easily why if $x > x_{i+4}$, then the value of the B-spline is zero. (It is assumed $x_{i+4}> x_{i+3} > \ldots > x_i$). But still, if $x < x_i$, then the value of $B_i(x)$ is not zero.