First-degree spline interpolation problem

Question

I have been spending hours working on a problem from Cheney & Kincaid on first-degree spline interpolation problem but am heading nowhere. Any explanation, pointers or hints from you would be bery much appreciated:

Defind a function $g$ by the equation:

$$g(x) = \begin{cases} 0, & \text{if $t_0$ $\le$ $x$ $\le$ 0} \\ x, & \text{if $0$ $\le$ $x$ $\le$ $t_n$} \end{cases}$$

Prove that every first-degree spline function that has knots $t_0, ..., t_n$ can be written in the form:

$$ax +b +\sum_{i=1}^{n-1} {c_i g(x-t_i)}$$

Thank you again for your time and help.

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EDIT: After many thoughts and in hindsight, I suspect that the question was not well written in the first place. I finally ended up with proving that the $S(x)$ is continuous on $x=0$ by stating that the limit of $S(x)$ from $0^+$ equals to that of from $0^-$ equals $S(0)$. I am sure it is the right solution but if it is not, at least it is not an irresponsible answer to a homework problem. Thank you.

Answer 1

Let $x_1<x_2<...<x_n$, $d_i=\frac{y_{i+1}-y_i}{x_{i+1}-x_i}$, $d_0=d_n=0$, and

$$s(x)=\cases{y_i+d_i(x-x_i), & $x_i\leq x<x_{i+1}$ \\y_n+d_n(x-x_n),& $x_n=x$}.$$ Now introducing the truncated power function $x_+=\max(x,0)=\frac{1}{2}(\vert x\vert+x)$ and the coefficients $c_i=\frac{1}{2}(d_i-d_{i-1})$ we can write it as a radial basis function sum with a low-degree polynomial in front:

$$s=y_1+\sum_{i=1}^{n}(d_i-d_{i-1})(x-x_i)_+ =\\ =y_1+\sum_{i=1}^{n}c_i\vert x-x_i\vert+\sum_{i=1}^{n}c_i(x-x_i)=\\ =\frac{y_1+y_n}{2}+\sum_{i=1}^{n}c_i\vert x-x_i\vert.$$

Answer 2

You know (probably) that the b-spline basis functions of degree 1 are little hat-shaped things that are non-zero over two knot spans. Take one of these hat functions, say the one that's non-zero on $[t_{i-1}, t_{i+1}]$, and try to write it as a linear combination of the functions $g(x - t_i)$. I think a linear combination with two terms will be sufficient. Draw some graphs to see what's going on.

If you can do this, then you're done, because you know that any spline of degree 1 can be written as a linear combination of the b-spline basis functions.