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The neutrix convolution product \(x_ -^{-r}\circledast x_ +^ \mu\). (English) Zbl 0765.46022

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

MSC:

46F10 Operations with distributions and generalized functions
44A35 Convolution as an integral transform
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