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Chebyshev system

  1. Jan 20, 2012 #1
    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 [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)

    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:

    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

    the determinant isn't equal to zero.

    Now I need to prove that there are no [itex]\beta_i [/itex] 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)
  2. jcsd
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