Wiener Tauberian theorems for \(L^1(K\setminus G/K)\)
Let \(G\) be a connected noncompact semisimple Lie group with finite center and \(K\) be a fixed maximal compact subgroup of \(G\). Let \(I(G)\) be the set of all complex valued spherical functions on \(G\): \[ I(G)= \{ f: f(k_1x k_2)= f(x), k_1,k_2 \in K, x\in G \}. \] Let \(dx\) denote a fixed Haar measure on \(G\) and let \(I_1(G)= I(G) \cap L^1(G)\). Then it is well known that \(I_1(G)\) is a commutative Banach algebra under convolution. The author proves a Wiener-Tauberian theorem for an arbitrary family of functions in \(I_1(G)\) with suitable conditions on the spherical Fourier transforms.
scientific article; zbMATH DE number 5542004