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.
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.