CHAPTER III - PARTICULAR TYPES OF GENERALIZED FUNCTIONS

Abstract

This chapter provides an overview of a few particular types of generalized functions. It discusses important functionals concentrated on smooth manifolds of lower dimension that is less than n imbedded in an n-dimensional space. The chapter highlights certain types of generalized functions associated with nondegenerate quadratic forms. It discusses specific regularizations of certain integrals without extension to the complex domain. The chapter describes the use of convergence from the complex domain. This complex extension method is simple. The chapter presents some useful information on delta functions associated with quadratic forms. It discusses quadratic forms whose coefficients are real and arbitrary homogeneous generalized functions of any degree in n dimensions. It reviews positive homogeneous functions of several independent variables as well. In the simplest case, that is, when G is a homogeneous nonnegative function, it is convenient for many reasons to introduce residues at the origin of a formally homogeneous function, and these residues characterize the singularity of this function at the origin in the same way that the residues of an analytic function characterize its isolated singularities.