Volterra systems and powers of Dirac delta impulses

Abstract

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

Convolution operators on some spaces of functions and distributions in the theory of circuits

“…The presented work was inspired by the talk [5] delivered by the first author during the conference on generalized functions in Bȩdlewo in 2007. We recall the results of the present authors given in [4] and in [16], extending them in section 7 by one of the results of the third author which are going to be published separately (see [4]). …”