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Approximation of Functions using the Sign Function

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that any function [itex] f(x)[/itex] can be approximated to any accuracy by a linear combination of sign functions as:

    [itex] f(x) ≈f(x_{0})+ \sum{[f(x_{i+1})-f(x_{i})]} \frac{1+ sgn(x -x_i)}{2}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    Looks like taylors theorem with a forward difference replaced with the derievative. It seems like the function only accepts sequential values approaching x from the left. That's about it. Anyone has any ideas?
     
    Last edited: Mar 25, 2012
  2. jcsd
  3. Mar 26, 2012 #2
    Doesn't look like Taylor series to me, nothing have been said about the difference between x_i and x_{i+1}. All I can see is that if {x_i} converges to x from the left, AND if f is left continuous, then the series converges to f(x). Not sure if this is necessary condition.
     
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