7. Basis functions

Just to be specific, let's consider splines with $n=4$. Make a nonparametric interpolating spline function $f _ 0 (t)$ with data values $1,0,0,0,0$, using the method of Section 5. Then make a similar function $f _ 1 (t)$ with data values $0,1,0,0,0$. Continuing this way you get functions $f _ 0 (t),\dots, f _ 4 (t)$, as shown in Figure . In terms of linear algebra, these functions are a basis for the vector space of all uniform relaxed cubic spline functions with $n=4$.

Figure: Basis functions for relaxed cubic splines, n=4

These basis functions have a practical use even for spline curves in higher dimensions with $n=4$: Instead of computing interpolating splines directly, as in Section 5, just compute these basis functions. Then, given data points $S _ 0,\dots, S _ 4$, just let $P(t) = f _ 0 (t) S _ 0 + \dots f _ 4 (t) S _ 4$. This should be reminiscent of the construction of Lagrange polynomials, and the reasoning to show that they do interpolate the data points is the same as in the case of Lagrange.

This method is valuable when there is a reason to do as much pre-computing as possible, so that there is nothing left to do with each new set of data points except to take a linear combination at each time. Two occasions when there is such a reason are these:

• You have many different curves to do with different data points but always for $n=4$.

• You need to interpolate in a space of high dimension, say for a complicated cartoon. This method avoids carrying the data through all the steps of row-reduction of the ``1 4 1'' matrix.