We consider a family of transforms which are generalizations of the Fo
urier transform on abelian groups. When the group under consideration
is Z(n), members of this family are characterized by having a matrix w
hose i,jth element is a(ij mod n), where a is any given vector. We add
ress the problem of when the inverse of a member of the family belongs
to the family. (C) Elsevier Science Inc., 1997.