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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