CYCLIC MAJORIZATION AND SMOOTHING OPERATORS

Recent work by M. L. Eaten and the present authors reviews and general izes work on majorization and group majorization. The standard materia l on majorization was extended from the symmetric group to more genera l groups in the important paper of Eaten and Perlman (1977). The prese nt paper studies one special nonreflection group, namely the cyclic gr oup on n elements. We say that a vector x cyclically majorizes a vecto r y, written y < Cx, if it lies in the convex hull of all vectors whic h can be obtained from x by cyclic permutation. The class of order-pre serving functions is studied, and the theory gives an ordering on the smoothness of periodic functions with possible application to time ser ies analysis and also an ordering of smoothing operators.