CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS

We present elementary proofs to some formulas given without proofs by K. A. West and J. McClellan in 1993, B. L. Evans and J. H. McClellan in 1994, and J. Cavicchi in 2002, for the calculation of the convolution integrals and sums of piecewise defined functions. Unlike “divide and conquer” strategy, these formulas are of the type “conquer what is divided”. Applications to differential equations, probability theory and linear discrete system theory are given. An example in connexion with the commutativity of the convolution product is also given. For completeness, in Annex we include a proof of the well-known elementary solution method for solving non-homogeneous linear differential equations with constant coefficients, used in the first application. The present work is part of a series of the author’s articles, some of which being published in this Journal, that present the products of convolution, both in discrete and continuous case, and some of their applications.