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A commutative neutrix convolution product of distributions. (English) Zbl 0821.46050

Summary: Let \(f\) and \(g\) be distributions in \({\mathcal D}'\) and let \[ f_ n(x)= f(x) \tau_ n(x),\quad g_ n(x)= g_ n(x) \tau_ n(x), \] where \(\tau_ n(x)\) is a certain function which converges to the identity function as \(n\) tends to infinity. Then the neutrix convolution product \(f\boxed{*} g\) is defined as the neutrix limit of the sequence \(\{f_ n * g_ n\}\), provided the limit \(h\) exists in the sense that \[ N- \lim_{n\to \infty} \langle f_ n * g_ n, \phi\rangle= \langle h, \phi\rangle \] for all \(\phi\) in \(\mathcal D\). The neutrix convolution products \(x^ \lambda_ - \boxed{*} x^ n_ +\) for \(\lambda, \mu,\lambda+ \mu\neq 0\), \(\pm 1,\pm 2,\dots\) and \(x^ \lambda_ -\boxed{*} x^ s_ +\) for \(\lambda\neq 0\), \(\pm 1, \pm 2,\dots\) and \(s= 0, 1, 2,\dots\) are evaluated, from which other neutrix convolution products are deduced.

MSC:

46F10 Operations with distributions and generalized functions
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