1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    [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]
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Chebyshev system
  1. Rössler system (Replies: 0)

  2. System of DEs (Replies: 0)

Loading...