×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] J.G. van der Corput: Introduction to the neutrix calculus. J. Analyse Math. 7 (1959-60), 291-398. · Zbl 0097.10503
[2] B. Fisher: Neutrices and the convolution of distributions. Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad 17 (1987), 119-135. · Zbl 0639.46041
[3] B. Fisher and Li Chen Kuan: A commutative neutrix convolution product of distributions. Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad (1) 23 (1993), 13-27. · Zbl 0821.46050
[4] B. Fisher, E. Özçaḡ and L. C. Kuan: A commutative neutrix convolution of distributions and exchange formula. Arch. Math. 28 (1992), 187-197. · Zbl 0788.46046
[5] I.M. Gel’fand and G.E. Shilov: Generalized functions, Vol. I. Academic Press, 1964.
[6] D.S. Jones: The convolution of generalized functions. Quart. J. Math. Oxford Ser. (2) 24 (1973), 145-163. · Zbl 0256.46054
[7] F. Treves: Topological vector spaces, distributions and kernels. Academic Press, 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.