Harmonic Analysis on Classical Groups

H.Weyl studied harmonic analysis on compact groups of finite di mension. He proved that an orthonormal system exists and that any continuous function on these groups can be approximated by some tinite linear combination of functions in this system. His research, however, seems to be too abstract to yield an explicit expression for the orthonormal system. Thus, we cannot talk about the form of the approximation, nor about its convergence. iO The simplest example of compact groups is {e }, on which there exists an orthonormal system inO { e }, n = 0, 1, 2 , ... , namely 1 J2" ." ." {I, for n = m; - e,n"e-1m"dO = 2n 0 0, for n =;6 m. The harmonic analysis on this compact group refers to the whole Fourier analysis. So far, extensive literature has been available on this topic. Its remarkable progress is evidenced by the great monograph of seven-hundred pages in two volumes written by A. Zygmund in 1959. iO An immediate extension for {e } is group U", which consists of all n X n square matrices U satisfying ufj' = I, where fj' denotes the conjugate transpose matrix of U. As for construction, there is a close relation between the group U and the group S03. Besides, 2 the application of U" has been found more and more important in physics.