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Derivative of best approximation

  1. Jun 10, 2015 #1
    Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less than n or basically anything satisfying the Haar condition).

    What can be said about how well the derivative, f'(x), is approximated by the derivative of the approximation?

    Thank you.
     
  2. jcsd
  3. Jun 10, 2015 #2

    Svein

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    Science Advisor

    Not very much. Take g(x)=f(x)+sin(1000000x)/1000 as an approximation to f(x). Then g'(x) = f'(x)+1000*cos(x).
     
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