Fourier Transformation of L2loc-functions

In this paper, we study the Fourier transformation of L2loc -functions and L2c-functions in order to investigate the natural statistical phenomena by using the theory of natural statistical physics. Thereby we prove the structure theorems of the image spaces FL2loc and FL2c. We study the convolution f ∗ g of a L2c-function f and a L2loc -function g. Further, we characterize the local Sobolev spaces and the space of solutions of Schrödinger equations. Here assume d ≥ 1. These results are the English version of Ito [17], chapter 5.