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Suppose I am trying to approximate a function which I do not know, but I can measure. Each measurement takes a lot of effort.

Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]##

Supose I know the expectation and variance of ##f(x)##.

Is there a way to compute the confidence in regression approximation as a function of sample size?

Is there a maximum number of useful measurements? For example, if I take 20 measurements between ##x=1## and ##x=2##, and 1 measurement at ##x=20##, another measurement at ##x=2## won't really tell me any new information right? So shouldnt there be a maximum number of measurements, or a minimum distance between measurements based upon the standard deviation of ##f(x)##?

Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]##

Supose I know the expectation and variance of ##f(x)##.

Is there a way to compute the confidence in regression approximation as a function of sample size?

Is there a maximum number of useful measurements? For example, if I take 20 measurements between ##x=1## and ##x=2##, and 1 measurement at ##x=20##, another measurement at ##x=2## won't really tell me any new information right? So shouldnt there be a maximum number of measurements, or a minimum distance between measurements based upon the standard deviation of ##f(x)##?

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