Results on the non-commutative product of distributions (Q2757058)

The authors study the non-commutative product of distributions \(x^r\ln^p|x|\) and \(x^{-r-1}\ln^q|x|\), for \(r= 0,\pm 1,\pm 2,\dots\) and \(p,q= 0,1,2,\dots\)\ . Such a non-commutative product is based on the following main definition.NEWLINENEWLINENEWLINEDefinition. Let \(f\) and \(g\) be distributions in \(D'\) and \(g_n(x)\) be defined by \(g_n(x)= (g* \delta_n)(x)\). We say that the product \(f\cdot g\) of \(f\) and \(g\) exists and is equal to the distribution \(h\) on the interval \((a,b)\) if NEWLINE\[NEWLINE\lim_{n\to\infty}\langle f(x) g_n(x),\varphi(x)\rangle= \langle h(x),\varphi(x)\rangleNEWLINE\]NEWLINE for all functions \(\varphi\in D\) with support contained in the interval \((a,b)\).NEWLINENEWLINENEWLINEFrom definition above, they obtain the following interesting theorems.NEWLINENEWLINENEWLINE(1) The product \((x^r\ln^p|x|)(x^{-r-1}\ln^q|x|)\) exists and is equal to \(x^{-1}\ln^{p+q}|x|\).NEWLINENEWLINENEWLINE(2) The product \((\text{sgn }x\cdot x^r\ln^p|x|)(\text{sgn }x\cdot x^{-r-1}\ln^q|x|)\) exists and is equal to \(x^{-1}\ln^{p+q}|x|\).