
Chebyshev system
it has been a while I was not post in this forum
I need a little help with this one
First, let me defined the chebyshev system.
A linearly independent set of continuous functions $\displaystyle \{u_0(x),u_1(x),\cdots, u_N(x)\}$ defined on $\displaystyle [a,b]$ is a Chebyshev system if for any $\displaystyle a\leq x_0<x_1<\cdots<x_N \leq b$ and $\displaystyle y_0,\cdots,y_N \in \mathbb{R}$, there is a unique linear combination $\displaystyle u(x)=\sum_{j=0}^N{a_ju_j(x)}$ satisfying $\displaystyle u(x_i)=y_i$ for $\displaystyle i=0,\cdots,N$
in $\displaystyle \mathbb{R}$, the set of $\displaystyle N$ continuous functions consisting of the powers of $\displaystyle x$ form a chebyshev system.
my question is why there is no Chebyshev system in $\displaystyle \mathbb{R}^s$ for $\displaystyle s \geq 2$ ?
thanks for your comment
br