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

C. K. Li and E. L. Koh

International Journal of Mathematics and Mathematical Sciences, 1998, vol. 21, 1-6

Abstract:

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].

Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:965895

DOI: 10.1155/S0161171298000969

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