A result on the composition of distributions
The authors define \[ f_n(x)= (f*\delta_n)(x)= \langle f(t); \delta_n(x- t)\rangle \] for \(n= 1,2,\dots\), where \(f\) is an arbitrary distribution in \({\mathcal D}'\), and \(\delta(x)\) the delta function. Using the definition \[ N-\lim_{n\to\infty} \int^\infty_{-\infty} F_n(f(x))\phi(x) dx= \langle h(x), \phi(x)\rangle, \] where \(\phi\) is a test function with compact support contained in \((a,b)\), \(f\) a locally summable function, and two theorems by the second author two lemmas are given to establish the main result that, the distribution \((x^r)^{-s}\) exists and \((x^r)^{-s}= x^{-rs}\), \(r\), \(s= 1,2,\dots\)\ . The proof is attractive and very neatly presented to arrive at a statement for the justification for any continuous function as \[ \lim_{n\to\infty} \Biggl|\int^\eta_{n^{-1/r}} x^{rs}[(x^r)^{-s}]_n\psi(x) dx\Biggr|= O(\eta) \] and \[ \lim_{n\to\infty} \Biggl|\int^{-n^{-1/r}}_\eta x^{rs}[(x^r)^{-s}]_n \psi(x) dx\Biggr|= O(\eta). \] Further, for any arbitrary function \(\phi(x)\) in \({\mathcal D}\) with compact support in \([-1,1]\), the authors employ Taylor's theorem with some obvious conditional steps to complete the proof of the theorem. A corollary defines the distribution \((x+i0)^{-s}\) to establish \[ (x^r+ i0)^{-s}= (x+ i0)^{-rs}. \]
scientific article; zbMATH DE number 1348063