Derivative of best approximation

[ekkilop]

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.

 

[Svein]

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

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).