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A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Authors:

Adem Kilicman

Abstract:

Let

$${tilde f}$$

,

$${tilde g}$$

be ultradistributions in $$mathcal{Z}{text{'}}$$

$$mathcal{Z}{text{'}}$$

and let

$$tilde fn = tilde f*delta n$$

and

$$tilde gn = tilde g*sigma n$$

where ${text{{ }}delta _n }$ is a sequence in $$mathcal{Z}$$

$$mathcal{Z}$$

which converges to the Dirac-delta function $$delta $$

$$delta $$

. Then the neutrix product $$tilde ftilde g$$

$$tilde ftilde g$$

is defined on the space of ultradistributions $$mathcal{Z}{text{'}}$$

$$mathcal{Z}{text{'}}$$

as the neutrix limit of the sequence $left{ {frac{1}{2}left( {tilde fntilde g + tilde ftilde gn} right)} right}$ provided the limit $${tilde h}$$

$${tilde h}$$

exist in the sense that

$mathop {{text{N - lim}}}limits_{n to infty } frac{1}{2}leftlangle {tilde f_n tilde g + tilde ftilde g_n ,psi } rightrangle = leftlangle {tilde h,psi } rightrangle$

for all

PSgr

in

$$mathcal{Z}$$

. We also prove that the neutrix convolution product $$fg$$

$$fg$$

exist in

$$mathcal{D}'$$

, if and only if the neutrix product $$tilde ftilde g$$

$$tilde ftilde g$$

exist in

$$mathcal{Z}{text{'}}$$

and the exchange formula $$F(fg) = tilde ftilde g$$

$$F(fg) = tilde ftilde g$$

is then satisfied.

Keywords:

distributions ultradistributions delta-function neutrix limit neutrix product neutrix convolution exchange formula

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