We shall define the spectral projection on the homogeneous tree $\mathfrak X$, which is an analogue of the one given by Bray for semisimple Lie groups. We shall prove the Paley--Wiener theorem for the spectral projection on $\mathfrak X$. As an application, we present an elementary proof of the Paley--Wiener theorem for the Helgason--Fourier transform on $\mathfrak X$, which was obtained by Cowling and Setti.
"An analogue of the spectral projection for homogeneous trees." Hiroshima Math. J. 43 (2) 207 - 221, July 2013. https://doi.org/10.32917/hmj/1372180512