Methods of constructing basis in solving inverse problems

Abstract

The paper suggests theoretical bases for building the complete systems of the Taylor-type basic elements with dense range of derivatives of positive, negative and fractional orders. This enables to explore the qualitative behavior of the solutions to differential and integral equations (direct and inverse problems), to process the experimentaldata, to identify the systems, to specify mathematical models of the objects and processes, to optimize the designing projects with specified characteristics, to prognosticate natural phenomena.