ABSTRACT
He has called it, circular average operator. It has been called in this chapter, generalized
translation operator associated with the Bessel operator. This operator is related to the product formula for the normalized Bessel function ja(^x)
studied in Chapter 1. More precisely we have the relation
In this chapter we study properties of the operators T^, x > 0, and we use them to define
the generalized convolution product *B of two regular functions f and g on [0, + °°[, by
where
We define also the generalized convolution product of complex bounded measures on
[0, + <»[ and of even distributions on JR. We establish many properties of this generalized convolution product *g on different
spaces of functions, on the space Mb([0, + °°[) and on spaces of even distributions. In
particular for all x > 0, y > 0, we have
where f is a continuous function on [0, + °°[, with compact support and 6^ the Dirac
measure at x. This relation and the properties of the generalized convolution product *B on
Mb([0, + °°[) imply that the pair ([0, + °°[, *B) is a hypergroup, called Bessel Kingmann
hypergroup. This is a particular case of the Chebli-Trimeche hypergroups (see [8] and [88]
Chap.2).
