(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the following system is a chebyshev system:

[tex]

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

[/tex]

2. Relevant equations

The set of [tex]

\{f_1, \cdots, f_n\}

[/tex] is a chebyshev system on [a,b] if the linear combination [tex]

\sum_{i=1}^n \alpha_i f_i

[/tex] has at most n-1 roots with [tex]

(\alpha_1, \cdots, \alpha_n) \neq (0, \cdots, 0)

[/tex]

3. The attempt at a solution

Look at the collection of functions without the logs then if it's evaluated for [itex] \{ x_i \}_1^n[/itex] then we get a van der monde matrix:

[tex]

\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}

[/tex]

the determinant isn't equal to zero.

Now I need to prove that there are no [itex]\beta_i [/itex] such that:

[tex]

\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)

[/tex]

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# Chebyshev system

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