Global interpolation or sinc interpolation is an ideal filter since its frequency response is a rect function. The impulse response of this filter is the sinc function (same as the coefficients of the interpolator).
Local interpolation methods are constructed by applying some window to the sinc function. For example, according to https://ccrma.stanford.edu/~jos/Interpolation/Relation_Lagrange_Interpolation_Windowed.html, for uniformly spaced samples and finite $N$ , Lagrange interpolaton is equivalent to windowed sinc interpolation using a binomial window. The impulse response of this filter are the coefficients of the interpolator.
I am interested to know what is the relationship between the impulse response, accuracy of the interpolation and frequency response of the filter/local interpolator, especifically for Lagrange interpolation. I cannot see this clearly.