I saw Spencer’s and Henderson’s weighted moving averages described in several places, but neither of the sources went into details. As I understand, both methods are designed in such way, so that if the underlying series could be described as polynomials up to 3-degree, then the smoothing procedure would lead to least possible distortion in the values (e.g. Hyndman, 2009, in short Moving averages paper). Could someone describe this in a greater detail? How the weights were derived? Why would we consider them as optimal? When shouldn't we use them?
You may want to take a look at "Statistics 153 (Time Series) : Lecture Three" - from Aditya Guntuboyina: lecture notes : it's pretty inclusive: https://www.stat.berkeley.edu/~aditya/Site/Statistics_153;_Spring_2012_files/Spring2012Statistics153LectureThree.pdf
The problem with this smoother is of course that it is time-symmetric / a-causal: if you are dealing with a real-time / online process, the shape of the curve will unfortunately be distorted. Just like a spline: looks pretty mathematically, but terrible once the data start moving.