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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 \(f\).

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