Let T be the unit circle in the complex plane. Let $/ be the Banach algebra of all complex valued continuous functions on T with absolutely convergent Fourier series. Norbert Wiener [10] proved that if any ƒ in sf is invertible in the ring of continuous functions on T, then \/f also is an element of sf . Paul Levy [7] generalized Wiener's result, showing that for each ƒ in sf and each complex a...

In this paper, we analyze the space D of distributions on the boundary Ω of a tree and its subspace B0, which was introduced in [Amer. J. Math. 124 (2002) 999–1043] in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if μ ∈ B0, then μ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω ,...

and some self-adjoint boundary condition at the origin. We assume that U is some bounded function for which HU has absolutely continuous spectrum. The presence of the absolutely continuous spectrum has direct consequences for the physical properties of the quantum particle described by the operator HU (see, e.g. [23, 2]). If we perturb this operator by some decaying potential V (x), the Weyl cr...