# When is an interpolating polynomial better than cubic spline interpolation?

The following plot is a slight variation of an example in a text book. The author used this example to illustrate that an interpolating polynomial over equally spaced samples has large oscillations near the ends of the interpolating interval. Of course cubic spline interpolation gives a good approximation over the whole interval. For years, I thought high order polynomial interpolation over equally spaced samples should be avoided for the reason illustrated here. However, I recently found many examples of bandlimited signals where a high order interpolating polynomial gives less approximation error than cubic-spline interpolation. It seems an Interpolating polynomial is more accurate over the entire interpolating interval when the sample rate is sufficiently high. This seems to hold when the samples are equally spaced with a sample rate at least 2.5 times greater than the Nyquist frequency of the signal. Furthermore, the advantage over cubic spline interpolation improves as (sample rate)/(Nyquist frequency) increasees.

As an example, I compare cubic-spline interpolation with an interpolating polynomial for a sine wave with a Nyquist frequency of 2 Hz, and a sample rate of 6.5 Hz. Between the sample points, tthe interpolating polynomial looks exactly the same as the actual signal. Below I compare the error in the two approximations. As with the first example, the polynomial interpolation does worst near the beginning and end of the sample interval. However, the interpolating polynomial has less error than a cubic spline over the whole sample interval. The interpolating polynomial also has less error when extrapolating over a small interval. Did I discover a well known fact? If so, where can I read about it? 