ON CERTAIN DISTRIBUTIONS MET IN SIGNAL THEORY AND OTHER DOMAINS OF SYSTEMS SCIENCE

摘要

The introduction in mathematics of concepts obtained as limits of already accepted ones has been a factor of progress. It has been the case with delta-pseudo-function (or delta-distribution). Here, we shall present what we call "epsilon-distribution", its theory and some applications. In a way epsilon-distribution is the anti-thesis of delta-distribution (or Dirac's delta). Heuristically it is a real pseudo-function ε(t) defined on R, "equal to zero everywhere but having an integral over R equal to one". It can be considered to have the limit of certain generative functions. It can also be presented, as usual in distribution theory, as a linear operator ε acting on functions u in such a way that it gives < ε,u > = m(u), the mean value of u on R. Applications are given in the fields of probability distributions, Schroedinger equation, heat equation, white noise, generalized harmonic analysis and pseudo-random functions.