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Journal of Lie Theory 21 (2011), No. 2, 427--456
Copyright Heldermann Verlag 2011

Nonabelian Harmonic Analysis and Functional Equations on Compact Groups

Jinpeng An
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R.China

Dilian Yang
Dept. of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4, Canada

Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation
f1(xy) + f2(yx) + f3(xy-1) + f4(y-1x) = f5(x)f6(y)
on arbitrary compact groups, where all fi's are unknown square integrable functions. It turns out that the structure of its general solution is analogous to that of linear differential equations. Consequently, various special cases of the above equation, in particular, the Wilson equation and the d'Alembert long equation, are solved on compact groups.

Keywords: Functional equation, Fourier transform, representation theory.

MSC: 39B52, 22C05, 43A30, 22E45

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