Cyclic majorization and smoothing operators Original Research Article

Recent work by M. L. Eaton and the present authors reviews and generalizes work on majorization and group majorization. The standard material on majorization was extended from the symmetric group to more general groups in the important paper of Eaton and Perlman (1977). The present paper studies one special nonreflection group, namely the cyclic group on n elements. We say that a vector x cyclically majorizes a vector y, written y < C x, if it lies in the convex hull of all vectors which can be obtained from x by cyclic permutation. The class of order-preserving functions is studied, and the theory gives an ordering on the smoothness of periodic functions with possible application to time series analysis and also an ordering of smoothing operators.