Given a function f ( x ) and two other functions g ( x ) and h ( x ), how would you determine if g ( x ) or h ( x ) is a better "approximation" of f ( x ).Does rigourous measure of approximation exist?

Question

Given a function   f  (  x  ) and two other functions   g  (  x  ) and   h  (  x  ), how would you determine if   g  (  x  ) or   h  (  x  ) is a better &quot;approximation&quot; of   f  (  x  ).Does rigourous measure of approximation exist?

Bruno Hughes · Accepted Answer

Yes, in fact there are many different measures of how &#039;good&#039; an approximation is; the notion of &#039;approximation&#039; depends on defining an underlying norm. For instance, let   C  [  0  ,  1  ] denote the class of continuous real-valued functions defined on the interval   [  0  ,  1  ]. We may define the supremum norm on this space by  ‖  f      ‖          ∞        :=  s  u  p  {      |    f  (  x  )      |    :  x  ∈  [  0  ,  1  ]  }  .With this norm in hand we can now give a rigorous definition of what &#039;close&#039; means in this normed space. Picking a very small   ϵ (say   0.0001) we might choose to say that functions   f and   ∈  C  [  0  ,  1  ] are &#039;close&#039; if  ‖  f  −  g      ‖          ∞        &amp;lt;  ϵ  .

Given a function f ( x ) and two other functions g ( x ) and h ( x ) ,

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