The theory of generalized functions

This paper gives an introductory account of the construction and properties of generalized functions f(x) of real variables x$_{1}$,x$_{2}$,$\cdots $,x$_{n}$. These are defined so as to ensure that (i) any generalized function f(x) possesses its full complement of generalized partial derivatives D$^{p}$f(x) of all orders; (ii) any convergent sequence of generalized functions {f$_{n}$(x)} has a generalized limit, f(x), which is also a generalized function; (iii) the derived sequence {D$^{p}$f$_{n}$(x)} converges to D$^{p}$f(x). The construction of these generalized functions ensures that any continuous function possesses derivatives which are generalized functions, so that the delta functions of Dirac are included in the theory. The representation of generalized functions by Fourier series and integrals is considered as an example of the simplicity and generality of the theory.