Nonabelian harmonic analysis and functional equations on compact groups
  Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is completely described. Consequently, several special cases of the above equation, in particular, the Wilson equation and the d'Alembert long equation, are solved on compact groups. 
Jinpeng An, Dilian Yang