Dirac delta function. Main properties and applications

Abstract

The article discusses some non-classical definitions of Dirac's delta function, its introduction into quantum mechanics and mathematics as an "exotic" discontinuous function, not strictly defined, but quite suitable for use in considering processes, some of whose characteristics change almost instantaneously. Its convenience lies in the fact that it itself, being "very discontinuous", helps to bring order in the research of precisely discontinuous functions. Some aspects of the strict definition of this function as a singular function - the Dirac distribution are also considered, its properties and application in many branches of mathematics and physics are determined. A comparative analysis of the classical and non-classical approach to the definition of the delta function was carried out. It is noted that the wider set of distribution functions includes the set of all previously considered classical functions and provides an opportunity to deepen the scope of application of mathematical analysis methods in the study of processes occurring in nature. Its importance in the application of Fourier spectral analysis and operational Laplace calculus for the study of discrete and continuous signals is indicated. Examples of the use of the delta function in mechanics, radio engineering, and probability theory are given. The WKS sampling theorem and its application to the discrete transmission of continuous non-periodic signals with a limited spectrum are considered. The technical capabilities of such transmission are specified.