Two Results on the Commutative Product of Distributions and Functions - Fisher, Brian (Sarajevo Journal of Mathematics, 2024)
Let f and g be distributions and let f_n=(f*δ_n x) and g_n =(g*δ _n)(x), where δ _n(x) is a certain sequence converging to the Dirac delta-function. The product f .g of f and g is defined to be the limit of the sequence {f_ng_n}, provided its limit h exists in the sense that _n → ∞ f_n (x) g_n (x),(x) =h(x), (x) for all functions in D. It is proved that align*(x|x| ^-r ^p |x|). ( |x| ^μ ^q|x|) &=x |x| ^-r +μ ^p+q|x| , (|x| ^-r ^p |x|). (x |x| ^μ ^q|x| ) &=x |x| ^-r +μ ^p+q|x| align* for -2<-r+μ< -1, r=1,2, … and p,q =0,1,2. … . 2000 Mathematics Subject Classification. 46F10