Pan-Linear Distribution And Differentiation

The distributions of L.Schwartz, the generalized distributions of Beurling, the ultredistributions of Roumieu and so on are continuous linear functionals defined on some suitable spaces of test functions. But they have a common characteristic: the condition of premise is too high. So we must improve it in order to fit the developement of Mathematics, Physics and Science.Recently, Professor Li Ronglu gives the definition of dissecting map, it includes all linear operators and various nonlinear maps. In this paper, the family of dissectting operators is used to develop the theory of distributions; it produces a new distribution-pan-linear distribution. The definition is that E∈{ K ( a ), K ,S} is a space of test functions and a function f :E→is called a pan-linear distribution if f is continuous and f∈F? ,U ( E, ). In fact every continuous linear functionals can be used to produce various pan-linear distributions, for example sin f (? ) , f∈K1 ( )′, e f( ? ) ? 1, f∈S ( )′and so on are continuous pan-linear distribution. So the family of pan-linear distribution also includes all linear distributions and various nonlinear functionals, it weakens the linear request of operators.First, the new basic principles of functional analysis are fundamental to the theory of pan-linear distributions in this paper. Our starting point is quite different from the hyperfunctions of M.Sato and the operational calculus of J.Mikusinski. On this base, the paper gives the definition of pan-linear distribution and some propositions.Subsequently, the operations of distribution are generalized to dissecting operators, the main perpose of this paper is to discuss the differentiations and multiplications for the dissecting functionals defined on the spaces of test functions. Meanwhile, it also discusses the convergence and gives the relationship between convergence and weak? convergence.Finally, this paper discusses the solution of the simplest equation y′= 0 in the pan-linear distributions. In the case of the usual distributions, it has solutions y = Constant only. However, we will show that the equation y′= 0 has extremely many solutions which are nonlinear dissecting functionals. This conclusion will be a nice fundamention of the solution of the equation y′= f.