The Neutrix Convolution of ultra-distributions Product And The Distributions on C^∞-Manifolds

Abstract

The absence of a general definition for convolutions and products of distribution is one of the issues with distribution theory. It is discovered in quantum theory and physics that certain convolutions and products such as 1/x+δ are in usea description of the term "product of distributions" and a list of sample product results using a particular delta sequence δ_n (x)=C_m n^m ρ(n^2 r^2 ) in an m-dimensional space. The Fourier transform is applied to D^' (m) and the exchange formula for defining ultradistribution convolutions in Z^' (m) in terms of products of distributions in D^' (m). We are going to demonstrate a theorem that says that for any items f ˜ and g ˜ in Z^' (m), the neutrix convolution f ˜⊗g ˜ exists in Z^' (m) if and only if the product f∘g exists in D^' (m). Some convolutional findings are derived using van der Corput's neutrix calculus. Let V'(M) be a smooth m-manifold M's space of distributions, each specified by an assemblage of 'compatible' ordinary distributions (components) displayed on the charts of some C^∞ on M. Drawing on van der Corput's concept of neutrix limitations, we expand the definition of the neutrix distribution product in this context. onto the space V'(M). We establish the existence of certain theorems regarding the neutrix distribution product in the space V'(M) under various assumptions on the neutrix product of the constituents.