AM  Vol.5 No.6 , April 2014
Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials
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 6×3 design with 3 treatments. In our example, the underlying group is the symmetric group S3.
Cite this paper: Zizler, P. (2014) Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials. Applied Mathematics, 5, 917-927. doi: 10.4236/am.2014.56087.

[1]   Davis, P.J. (1979) Circulant Matrices. John Wiley and Sons.

[2]   Dummit, D.S. and Foote, R.N. (1999) Abstract Algebra. John Wiley and Sons.

[3]   Morrison, K.E. (1998) A Generalization of Circulant Matrices for Non-Abelian Groups. Private Communication.

[4]   Zizler, P. (2013) On Spectral Properties of Group Circulant Matrices. Pan American Mathematical Journal, 23, 1-23.

[5]   Foote, R., Mirchandani, G., Rockmore, D., Healy, D. and Olson, T. (2000) A Wreath Product Group Approach to Signal and Image Processing: Part I—Multiresolution Analysis. IEEE Transaction in Signal Processing, 48, 102-132.

[6]   Healy, D., Mirchandani, G., Olson, T. and Rockmore, D. (1996) Wreath Products for Image Processing. Proceedings of the ICASSP, 6, 3582-3586.

[7]   Malm, E.J. (2005) Decimation-in-frequency Fast Fourier Transforms for the Symmetric Group. Thesis, Department of Mathematics, Harvey Mudd College.

[8]   Maslen, D. and Rockmore, D. (1997) Generalized FFTs—A Survey of Some Recent Results. In: Finkelstein, L. and Kantor, W., Eds., Proceedings of the DIMACS Workshop on Groups and Computation, 183-237.

[9]   Rockmore, D.N. (2003) Recent Progress and Applications in Group FFTs. NATO Advanced Study Institute on Computational Noncommutative Algebra and Applications.

[10]   Sagan, B.E. (1991) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Wadsworth & Brooks/Cole, Pacific Grove.

[11]   Terras, A. (1999) Fourier Analysis on Finite Groups and Applications. Cambridge.

[12]   Stankovic, R.S., Moraga, C. and Astola, J.T. (2005) Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design. IEEE Press, John Wiley and Sons.

[13]   Karpovsky, M.G. (1977) Fast Fourier Transforms on finite Non-Abelian Groups. IEEE Transaction on Computers, C-26, 1028-1030.

[14]   Senn, S. (2002) Cross-Over Trials in Clinical Research. 2nd Edition, Wiley.

[15]   Chow, S.-C. and Liu, J.-P. (2013) Design and Analysis of Clinical Trials: Concepts and Methodologies. Wiley.