Spectral analysis on homogeneous trees (Q1271892)

A homogeneous tree \(T\) of degree \(q+1\) is considered with a fixed origin \(o\). Let \(f\) be a complex valued function on the vertices of \(T\) and denote the arithmetic mean of \(f\) over the nearest neighbors of the vertex \(x\) by \(Pf(x)\). Then the Laplacian of \(T\) is \(P - I\), where \(I\) denotes the identity. For \(z\in\mathbb{C}\setminus\{0\}\) the authors construct an operator \(H_z\) that maps functions harmonic on \(T\) bijectively to \((z-1)\)-eigenfunctions of the Laplacian. Their ansatz is \[ H_zf(v_m) = z^m \sum_{k=0}^m \xi_k f(v_{m-k}) \text{ and } PH_zf(v_m) = z H_zf(v_m) + z^{m +1} (P - I)f(v_m), \] where \(v_m\) is a vertex of (shortest path) distance \(m\) from \(o\) and \(\{o, v_1, \ldots, v_m\}\) the (unique) minimizing path. The coefficients \((\xi_k)_k\) are then determined recursively. The operator \(H_z\) is inverted and compared to a similar operator of \textit{A. M. Mantero} and \textit{A. Zappa} [J. Funct. Anal. 51, 372-399 (1983; Zbl 0532.43006)]. Finally, a general function is represented as an integral of \((z-1)\)-eigenfunctions with respect to a distribution.