Abstract :
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.
