## A comparison on the commutative neutrix convolution of distributions and the exchange formula.(English)Zbl 1079.46514

Summary: Let $$\tilde f$$, $$\tilde g$$ be ultradistributions in $$\mathcal Z'$$ and let $$\tilde f_n = \tilde f * \delta _n$$ and $$\tilde g_n = \tilde g * \sigma _n$$ where $$\{\delta _n \}$$ is a sequence in $$\mathcal Z$$ which converges to the Dirac-delta function $$\delta$$. Then the neutrix product $$\tilde f \diamond \tilde g$$ is defined on the space of ultradistributions $$\mathcal Z'$$ as the neutrix limit of the sequence $$\{{1 \over 2}(\tilde f_n \tilde g + \tilde f \tilde g_n)\}$$, provided that the limit $$\tilde h$$ exists in the sense that $\text{N-}\!\!\!\lim _{n\to \infty }{1 \over 2} \langle \tilde f_n \tilde g +\tilde f \tilde g_n, \psi \rangle = \langle \tilde h, \psi \rangle$ for all $$\psi$$ in $$\mathcal Z$$. We also prove that the neutrix convolution product $$f \diamondsuit g$$ exist in $$\mathcal D'$$ if and only if the neutrix product $$\tilde f \diamond \tilde g$$ exist in $$\mathcal Z'$$ and the exchange formula $$F(f \diamondsuit g) = \tilde f \diamond \tilde g$$ is then satisfied.

### MSC:

 46F10 Operations with distributions and generalized functions
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### References:

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