A basis for space of spline of degree k

Question

how can i proof space of spline of degree k on $[x_0,x_n]$ with knots: $$x_0<x_1<\cdots<x_{n-1}<x_n,$$ have a basis like this: $$\{1,x,x^2,\ldots,x^k,(x-x_1)^k_+,(x-x_2)^k_+,\ldots,(x-x_{n-1})^k_+\}.$$

remark:

$$(x-x_i)^k_+=\left\{\begin{array}{2}(x-x_i)^k& x>x_i\\0& x\leq x_i\end{array}\right.$$

Answer 1

Kincaid D., Cheney W. Numerical analysis, Chapter 6:

A natural spline of degree $2m+1$ is a function $s\in C^{2m}(\mathbb{R})$ that reduces to a polynomial of degree $\leq2m+1$ in each inner interval and to a polynomial of degree at most $m$ in $(-\infty,t_{1})$ and $(t_{n},\infty)$.

Theorem. Every natural spline can be represented using truncated power functions as follows $$ s(x)=\sum_{k=0}^{m}\alpha_{k}x^{k}+\sum_{i=1}^{n}\beta_{i}(x-x_{i})_{+}^{2m+1} $$ with the coefficient conditions, $$ \sum_{i=1}^{n}\beta_{i}x_{i}^{k}=0,\qquad k=0,1,\ldots m $$

Answer 2

Thanks for your answer @rych. Consider spline $s$ of degree $k=2m+1$ for nodes $x_0<\cdots<x_n$ in whole line $\mathbb{R}$ instead of $[x_0,x_n]$.

Definition of spline: \begin{array}{l} 1.\qquad s_i(x)=s(x)|_{[x_i,x_{i+1}]}\in\mathbb{P}_{2m+1},\qquad i=0,1,\ldots,n-1,\\ 2.\qquad s\in\mathbb{C}^{2m}(\mathbb{R}). \end{array} Definition of natural spline based on "Numerical Mathematics" by A. Quarteroni, page 357 : $$s^{(i)}(x_0)=s^{(i)}(x_n)=0,\quad i=m+1,\ldots,2m.\qquad (*)$$ $s_0\in\mathbb{P}_{2m+1}$ and $s_0\in\mathbb{C}^{2m}(\mathbb{R})$.

Consider taylor expansion of $s_0$ at $x_0$: $$\begin{align} s_0(x)=&\sum_{i=0}^{2m}{\frac{s_0^{(i)}(x_0)}{i!}(x-x_0)^i}+\left[\frac{s^{(2m+1)}(x_0+)-s^{(2m+1)}(x_0-)}{(2m+1)!}\right](x-x_0)^{2m+1}\\ \stackrel{(*)}{=}& p_m(x)+b_0(x-x_0)^{2m+1},\qquad x\in[x_0,x_1]. \end{align}$$. How can i proof $s|_{(-\infty,x_0],[x_n,\infty)}\in\mathbb{P}_m$?