Convolution, filtering, linear systems, the Wiener-Khinchin theorem: generalizations

A simple formulation is given for generating convolution theorems in any representation. Using this method we obtain the convolution theorem for the scale representation. We generalize the concept of invariance to any basis set and devise a method for handling linear invariant systems for arbitrary quantities. The Wiener-Khinchin theorem is generalized to arbitrary power energy densities. Also, we show how standard probability theory can be formulated in terms of signals.