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Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f $ called $\tilde{f}$ using the given data points.

When such aproximation is called interpolation? should the approximation vanish on the given data points in order to be considered as an interpolation ( i.e. $\tilde{f}(x_i)=y_i$ for all $i$) ? Thank you in advance.

gt6989b
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Nizar
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  • As Ross Millikan said, there is a difference in terms. If you want to match the data exactly, it is called interpolation, if not -- approximation – gt6989b Feb 08 '16 at 15:59
  • I will add this comment for users checking the same question. http://math.stackexchange.com/questions/65532/interpolation-extrapolation-and-approximations-rigorously – Nizar Feb 08 '16 at 16:42

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Yes, an interpolating function should pass exactly through all the data points. You need enough adjustable parameters to make this happen. It can also be useful to find an approximating function that has fewer parameters and does not pass exactly through the points. If there is noise in your data, some forms of interpolating function, like a high degree polynomial, will wiggle a lot between the points.

Ross Millikan
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  • Thank you for your answer, but would you please add a reference to support your answer as "Yes". I mean on what did you base your answer? Is there a clear definition ? – Nizar Feb 08 '16 at 16:05
  • If you look at the Wikipedia page it shows various interpolation techniques, all of which produce functions that go exactly through the points. – Ross Millikan Feb 08 '16 at 17:57