The neutrix convolution product in Z ′ ( m ) and the exchange formula

One of the problems in distribution theory is the lack of definition for convolutions and products of distribution in general. In quantum theory and physics (see e.g. [1] and [2]), one finds that some convolutions and products such as 1 x ⋅ δ are in use. In [3], a definition for product of distributions and some results of products are given using a specific delta sequence δ n ( x ) = C m n m ρ ( n 2 r 2 ) in an m -dimensional space. In this paper, we use the Fourier transform on D ′ ( m ) and the exchange formula to define convolutions of ultradistributions in Z ′ ( m ) in terms of products of distributions in D ′ ( m ) . We prove a theorem which states that for arbitrary elements 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 results of convolutions are obtained by employing the neutrix calculus given by van der Corput [4].