How signal moments work? We know that moments extract important features of a signal that has no redundancy. For example Zernike moment gives us rotation invariant data of a signal like image or Fourier gives us frequency of a signal to us. 


But what is the theory behind it? 
How a moment performs this? 
How basis function of a moment is designed?


I think it has roots in many different math theory but I can't find out a good source to know.
I dont think there is a common framework for defining all those operators. Each of them were developed by different people, at several moments of the history. They are being integrated day by day onto a common discipline called DSP.  

As clarification, an operator (like a moment or any other) and the "redundancy" of a signal are very different and unrelated concepts. 

IF you want a deeper answer, list all the moments you are considering, and all the different math theories you are reviewing for each of them.