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Extensions of two theorems on the neutrix convolution product of distributions. (English) Zbl 0743.46027

The paper is clearly developed and very well written. It begins with the classical convolution of two distributions, \(f\) and \(g\) with either having bounded support or both having bounded support on the same side. Then this gives the well defined convolution product, \(\langle(f*g)(x),\phi\rangle =\langle f(y),\langle g(x),\phi(x+y)\rangle\rangle\) for all \(\phi\in{\mathcal D}\). We loose the commutativity of the convolution when we extend it to neutrix convolution but relax the support constraints. The neutrix convolution product, \(f(*)g\), for two distributions, \(f\) and \(g\) is defined as the neutrix limit of the sequence, \(\{f_ n*g\}\), where \(\{f_ n\}\) is a certain sequence converging of \(f\). Two principle results already proven for \(\lambda>-1\), \(\lambda\not\in\mathbb{N}_ 0\) and \(s\in I\) are extended to the situation where \(\lambda\not\in I\) and \(I\) are all integers. The two results are \[ x^ \lambda (*)x_ +^{s-\lambda}=(-1)^{s+1}\cdot B(- s-1,s+1-\lambda)x^{s+1}+{(-1)^{s+1}(\lambda)_{s+1} \over (s+1)!} [\pi \cot(\pi\lambda)x_ +^{s-\lambda}-x^{s+1}\ln| x|] \] and \[ x_ -^ \lambda (*)x_ +^{-s-\lambda}={\pi \cot(\pi\lambda)x \over (-1-\lambda)_{s+1}}\delta^{(s-2)}(x)-{(-1)^ s(s-2)! \over (-1- \lambda)_{s-1}}x^{-s+1}. \] The proofs are very well done.

MSC:

46F10 Operations with distributions and generalized functions
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