Why is the use of high order polynomials for regression discouraged? I've read many times on this site that high order polynomials (generally more than third) shouldn't be used in linear regression, unless there is a substantial justification to do so.
I understand the issues about extrapolation (and prediction at the boundaries).
Since extrapolation isn't important to me...

*

*Are high order polynomials also a bad way of approximating the underlying function within the range of the data points? (i.e. interpolation)

*If so, what problems are arising?

I don't mind being redirected to a good book or paper about this.
Thanks.

Runge's phenomenon can lead to high-degree polynomials being much wigglier than the variation actually suggested by the data. An appeal of splines as a substitute for high-degree polynomials, particularly natural splines, is to allow nonmonotonicity and varying slopes without varying too wildly. I would be hard-pressed to find an example of real data for which a fourth- or higher-degree polynomial seems a more reasonable choice than a spline.
