Motivation and main types of functional transformation

The use of so-called functional transformations (because they apply to possibly generalized functions) is already quite old, and has come into common use for a number of different reasons.

— We want to store information, or reconstitute a signal, or show particular properties of a signal, or of a set of measurements.

— We want to perform some quick numerical calculations.

— We want to transform a complicated problem into a simpler one, if possible solvable. Let's be a little more precise.

Suppose we have to solve a stationary problem of the type :

with f given in a (functional) space X, A a (linear) operator in...