The convolution of functions and distributions

The non-commutative convolution f ∗g of two distributions f and g in D is defined to be the limit of the sequence {(f τn) ∗ g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that |x|λ ∗ sgn x|x|μ = 2sin(λπ/2) cos(μπ/2) sin[(λ+μ)π/2] B(λ+1,μ +1) sgn x|x|λ+μ+1, for −1<λ+μ<0 and λ,μ =−1,−2, . . ., where B denotes the Beta function.  2005 Elsevier Inc. All rights reserved.