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What exactly does higher-order continuity mean because low-degree polynomials are still infinitely continuous/differentiable.


Why do the higher degree polynomials have these artifacts at the endpoints?


This looks similar to how the Fourier series of a square wave behaves - as you get closer and closer to the discontinuity, the Fourier series starts oscillating more and more due to the high frequency functions that are needed to model the discontinuity


Do we always need to worry as it gets worse far away from 0? Are there times when we only care about approximating near a particular point?


Can we extend the range of approximation, so that the endpoints are less important?


Is there a way to use the higher order polynomial for the parts of the interval that it approximates well and then interpolate it with lower order polynomials near the end points to avoid the weird oscillation?


Is there maybe a way of using low pass filters of some sort close to the endpoints so that way we can filter out any extraneous/excessive values at these areas?


How would we determine when our polynomial degree is too great in the context of live programs/applications (without obvious visualization/comparison such as above^).


Overfitting is also a common issue in machine learning, which reminds me of the similarity of these two problems. Can neural networks help interpolation in this case?