No abstract

One-argument and two-argument operations of distributional composition are studied for real-valued functions and distributions on R. Several results are proved on the existence of the distributional composition [g] • [f ] for functions (and on the consistency with the usual functional composition g • f ) as well as for distributions, e.g. in case g is a continuous function and f is a measure or, more generally, a semi-regular (almost regular) distribution, i.e. a distribution such that the Łojasiewicz value f Ł (a) of f at a point a exists for almost all a on R and the mapping f R : x → f Ł (x) is a measurable (locally integrable) function. Several formulas involved with the Dirac delta distribution δ are given.

The theory of linear and nonlinear systems, used in electronics, signal processing and telecommunications, is mathematically clarified by extending the operators considered in the theory from spaces of functions to the space of distributions. In case of nonlinear systems, expressed by infinite sums of homogeneous systems of degree k called the Taylor and Volterra series, the extension requires, in particular, a justification of the operation of the kth power of the Dirac delta distribution. This is accomplished due to the notion of neutrix product of distributions meant in a wider sense than it was considered before.Keywords: linear and nonlinear system (circuit); Taylor and Volterra series; powers of the Dirac delta impulse; product of distributions; neutrix product of distributions