So I was trying to understand what these spline functions actually mean, but I am stuck. What I came up with was that between 2 specific nodes xi-1 and xi+1, depending on where x is, the spline functions give us a fraction of an interval.
Thank you for your help. I want to understand how the linear spline interpolation works, but this is different from what I found on the internet, where I didn't find this specific form.
Thank you
These are the b-spline basis functions of degree 1. Each of them is a “hat” or “tent” shape. The $i$-th function $S_i$ has the value 1 at $x_i$, and then it’s value descends linearly to zero on either side (at $x_{i-1}$ and $x_{i+1}$).
By taking linear combinations of these basis functions, you can get any piecewise linear function. In fact, the piecewise linear function that has the value $y_i$ at $x=x_i$ is just $$ S(x) = \sum y_iS_i(x) $$ B-spline basis functions are often defined recursively, via the deBoor-Cox algorithm: the degree 2 one’s are defined in terms of the degree 1 one’s, the degree 3 ones are defined using the degree 2 ones, and so on. Look up b-spline basis functions, or look here or here.