Real analytic generalized functions (Q1006427)

Let \(\omega\) denote an open set in \(\mathbb R^d\) and \(x_0\) be an element of \(\omega\); let \({\mathcal G}(\Omega)\), \(\Omega\in C^p\), be the generalized function algebra (Colombeau algebra). A generalized function \(g\in{\mathcal G}(\Omega)\) is said to be real analytic at \(x_0\) if there is an open ball \(B=B(x_0,r)\subset\omega\) containing \(x_0\) and \((g_\varepsilon)_\varepsilon\in{\mathcal E}_M(B)\) such that (i) \(f|B=[(g_\varepsilon)_\varepsilon]\) in \({\mathcal G}(B)\); (ii) \((\exists\eta > 0)(\exists\varepsilon_0\in(0,1))(\sup_{x\in B}|g_\varepsilon^{(\alpha)} (x)|\leq \eta^{|\alpha|+1}\alpha!\varepsilon^{-a}\), \(a>0\), \(0<\varepsilon< \varepsilon_0\), \(\alpha\in \mathbb N^d\). It is said that a generalized function \(g\) is real analytic in \(\omega\) if \(g\) is real analytic at each point of \(\omega\). It is also proved that the strong associativity of a distribution and a generalized function is a necessary and sufficient condition for the equality of their analytic wave front sets.