# Chebyshev system

1. Jan 20, 2012

### dirk_mec1

1. The problem statement, all variables and given/known data
Show that the following system is a chebyshev system:

$$\{1, x, x^2, \cdots, x^n, x \log(x), \cdots, x^m \log(x)\}, n \geq m$$

2. Relevant equations
The set of $$\{f_1, \cdots, f_n\}$$ is a chebyshev system on [a,b] if the linear combination $$\sum_{i=1}^n \alpha_i f_i$$ has at most n-1 roots with $$(\alpha_1, \cdots, \alpha_n) \neq (0, \cdots, 0)$$

3. The attempt at a solution
Look at the collection of functions without the logs then if it's evaluated for $\{ x_i \}_1^n$ then we get a van der monde matrix:

$$\begin{pmatrix} 1 & 1 & \cdots & 1 \\ x_1 & x_1^2 & \cdots & x_1^n \\ \vdots & \vdots & \ddots & \vdots \\ x_{n+1} & \cdots & \cdots & x_{n+1}^n \end{pmatrix}$$

the determinant isn't equal to zero.

Now I need to prove that there are no $\beta_i$ such that:

$$\log(x) (\gamma_1 x + \gamma_2 x^2 +...+ \gamma_m x^m) = \beta_1 x + \beta_2 x^2 +...+\beta_m x^m + \beta_{m+1} x^{m+1}+...+\beta_n x^n\ \forall\ x \in (0,1)$$