Some Results on the Composition of Distributions - Ege, Inci (Sarajevo Journal of Mathematics, 2024)
Let $F$ be a distribution, $f$ be a locally summable function and $F_{n} = (F * δ_{n})(x)$, where $δ_{n}(x)$ is a certain sequence converging to the Dirac delta-function $δ(x)$. The distribution $F(f)$ is defined as the neutrix limit of the sequence $\{ F_n(f) \}$, provided its limit $h$ exists in the sense that $$ \Nn \int_{-∞}^{∞} F_{n}(f(x))φ (x) dx = \la h(x) , φ (x) \ra $$ for all test functions $φ(x)$ in ${\mathcal D}(a,b)$. The composition of the distributions $x_{-}^{-s}\ln x_{-}$ and $x_{+}^{r}$ is evaluated for $r=0,1, …$ and the composition of the distributions $x_{-}^{-s}\ln^{m} x_{-}$ and $H(x)$ is evaluated for $s,m=1,2,…\,$. 2000 Mathematics Subject Classification. 46F10