Developments of the theory of generalized functions or distributions – A vision of Paul Dirac

This paper deals with six major developments of the theory of generalized functions or distributions as a whole new branch of modern analysis. These include (i) The Dirac delta function as the limit of sequences of ordinary functions, (ii) Schwartz´s new theory of distributions based on test functions, (iii) Temple–Lighthill´s simpler analytical approach to generalized functions based on good functions, (iv) Mikusinski´s algebraic approach to generalized functions, (v) Cauchy´s representation of distributions by analytic functions, and (vi) Sato´s less abstract and computationally more effective approach to generalized functions (or hyperfunctions) based on complex variables. Included are basic features, properties and examples of generalized functions including the Dirac delta function and the Heaviside function. Some additional properties of convolution are discussed in the end.