A Paley-Wiener theorem for central functions on compact Lie groups (Q2759759)

The author begins with a Paley-Wiener type theorem for functions on the \(n\)-torus. This result is then used to derive an analogous result for central functions on a compact semisimple Lie group \(U\). Fix a maximal torus \(T\subset U\) and let \(0<R<2\pi/|\delta|\), where \(\delta\) is a root of maximal length, and set \({\mathbf t}= \text{Lie}(T)\). The statement of the result is: A central function \(f\) in \(C^\infty(U)\) has support in the ball of radius \(R\) if and only if the Fourier transform \(\widehat f(\mu)\) extends to a holomorphic function on \(({\mathbf t}^*)^c\) of exponential type \(R\) satisfying the skew condition \(\widehat f(\mu)=(\det s)\widehat f(s(\mu+\rho)-\rho)\).NEWLINENEWLINEFor the entire collection see [Zbl 0970.00022].