Relationship between interpolation accuracy, impulse response and frequency response
any interpolation polynomial that goes through the two adjacent samples to the left and right of the area being interpolated will result in an effective impulse response that is a piecewise-polynomial function going through zero at all integers except zero (like the sinc function does).

e.g. linear interpolation:

 

third-order lagrange:



third-order hermite:



note all of these effective impulse responses go through the $(x,y)$ points of $(0,1)$ and $(n,0)$ for any other integer $n \ne 0$, just like the sinc function does. 



so you can represent that interpolation function as something times a sinc function and the "something" can be obtained by dividing that interpolation function, point-by-point, with the sinc function.  the only division by zero is when the numerator is also zero and that can be, what we like to call in the hand-waving department, a "removeable singularity".  (maybe we can find a decent limit at the $\tfrac00$ divisions.)

any interpolation that forces the smooth interpolated function to go through the discrete points can always be described as a windowed-sinc function.  all you have to do is divide to get the effective window.