Commutative convolution of functions and distributions

Abstract

The commutative convolution f * g of two distributions f and g in D' is defined as the limit of the sequence {(f tau(n)) * (g tau(n))}, provided the limit exists, where {tau(n)} is a certain sequence of functions tn in D converging to 1. It is proved that

|x|(lambda) * (sgn x|x|(-lambda-1)) = pi[cot (pi lambda) - cosec(pi lambda)] sgn x|x|(0),

for lambda not equal 0, +/- 1, +/- 2, ... , where B denotes the Beta function