Rate of convergence for functions

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I am not very familiar with terms from numerical analysis, thus I do understand the definition for convergence rate from http://en.wikipedia.org/wiki/Rate_of_convergence" [Broken]. Still, here the definition appears only for sequences.

Which is the definition for rate of convergence for functions? For instance: for I closed and bounded set, and for O discrete set, a function F:I->O, x-> F(x) is approximated by a set of functions G: I X R+ ->O , (x,p(k)) ->G(x,p(k)), where p: R+->R+, k ->p(k) is a monotonic decreasing function, and R+ denotes the positive real numbers. The set of functions G converge towards F, i.e. lim_{k->0} G(x,p(k))=F(x). Which is the convergence rate for G?
Any idea on how rate of convergence would be defined in this way? What does it mean if the rate of convergence is infinity in this case?

Thank you very much for your help.
 
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  • #2
mathman
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The rate of convergence would be how fast |G(x,p(k))-F(x)| -> 0 as a function of k. Since we are dealing with functions, you need to define a norm, for example max over all x.
 

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