Math. Z. (2011) 269:555–586 DOI 10.1007/s00209-010-0750-1 Mathematische Zeitschrift Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type Sanjoy Pusti · Swagato K. Ray · Rudra P. Sarkar Received: 19 March 2010 / Accepted: 8 June 2010 / Published online: 2 July 2010 © Springer-Verlag 2010 Abstract We will show that an uniform treatment yields Wiener–Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/ K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/ K . The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze–Stein phenomenon for noncompact semisimple Lie groups. Keywords Lorentz spaces · Spinor · Wiener–Tauberian theorem Mathematics Subject Classification (2000) Primary 43A85; Secondary 22E30 1 Introduction A classical theorem of Norbert Wiener [51] states that if the Fourier transform of a function f ∈ L 1 (R) has no real zeros then the finite linear combinations of translations f (x - a) of f with complex coefficients form a dense subspace in L 1 (R), equivalently, span{g ∗ f | g ∈ L 1 (R)} is dense in L 1 (R). This theorem which is well known as the Wiener–Tauberian Theo- rem (WTT), was generalized for abelian locally compact groups where the hypothesis is on a Haar integrable function which has nonvanishing Fourier transform on all unitary characters. S. Pusti (B ) · R. P. Sarkar Stat-Math Unit, Indian Statistical Institute, 203 B. T. Rd., Calcutta 700108, India e-mail: spusti_r@isical.ac.in R. P. Sarkar e-mail: rudra@isical.ac.in S. K. Ray Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India e-mail: skray@iitk.ac.in 123