Norm Convergence in the Space of Hyper-functions

Abstract

Classical Mathematics is not sufficient to justify some functions like Dirac’s delta function, Heaviside’s unit step function in the mathematical modelling of some physical problems. Mikio Sato introduced the concept of hyperfunction to explain such situations. He gave a new generalisation to such functions using the theory of complex analysis. Hyperfunctions have many applications in the field of differential equations that are related with the physical problems involving Heat equation, wave equation etc. Urs Graf applied various transforms to hyperfunctions. With the help of these transforms he solved differential equations in terms of hyperfunctions. In this paper we defined a norm to a subclass of the linear space of hyperfunctions. The completeness and separability properties of this subfamily of hyperfunctions are established in this paper. Hyperfunctions of bounded exponential growth with compact support are mainly considering here.We have developed the results using the defining function of the hyperfunction. Hence we give a normed space approach to the subfamily of hyperfunctions having bounded exponential growth with compact support.
Mathematics Subject Classification: 46F15, 46BXX