TITLE:
Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials
AUTHORS:
Peter Zizler
KEYWORDS:
Non-Abelian Fourier Transform, Group Algebra, Irreducible Representation, Irreducible Character, G-Circulant Matrix, G-Decorrelated Decomposition, Crossover Designs in Clinical Trials
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.6,
April
8,
2014
ABSTRACT: Let G be a non-abelian
group and let l2(G) be a
finite dimensional Hilbert space of all complex valued functions for which the
elements of G form the (standard) orthonormal basis. In
our paper we prove results concerning G-decorrelated
decompositions of functions in l2(G). These G-decorrelated
decompositions are obtained using the G-convolution either by
the irreducible characters of the group G or by an orthogonal
projection onto the matrix entries of the irreducible representations of the
group G. Applications of these G-decorrelated
decompositions are given to crossover designs in clinical trials, in particular
the William’s 6×3design with 3 treatments. In our example, the
underlying group is the symmetric group S3.