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Mikusiński type products of distributions in Colombeau algebra. (English) Zbl 1021.46032
The well-known equation \[ x^{-1}\cdot x^{-1}-\pi^{-2}\delta(x)\cdot \delta(x)= x^{-2},x\in \mathbb{R} \] in which the “product” of distributions appeared, is generalized for the Colombeau algebra \(G_\tau\) of tempered generalized functions. This algebra \(G_\tau\) contains the space of tempered distributions, canonically embedded by the mapping: \(S'\to G_\tau: u\to \{ \widetilde u(w,x)= (u*\check w)(x)\}\), \(w\in A_0\), where \(A_0=\{w(x)\in D(\mathbb{R}); \int_{\mathbb{R}} w(x)dx=1\}\).
The main result is the following: For any \(p,q\in \mathbb{N}_0\) the embeddings in \(G_\tau\) of distributions \((x^{-1})^{(p)}\) and \(\delta^{(q)} (x)\), \(x\in \mathbb{R}\), satisfy \[ \bigl((\widetilde{x^{-1}})^{(p)} \bigr)\bigl( \widetilde {\delta^{(q)}} (x)\bigr) +\bigl((\widetilde {x^{-1}})^{(q)} \bigr) \bigl(\widetilde {\delta^{(p)}}(x) \bigr)\approx c_{p,q}\delta^{(p+q+1)} (x), \] where \(c_{p,q}= -p!q!/(p+q+1)\) and \(\approx\) for \(f\in G\) and \(u\in S'\), \(f \approx u\), means that \(u\) is “associated” to \(f\).

MSC:
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
44A40 Calculus of Mikusiński and other operational calculi
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