HARMONIC-ANALYSIS OF SPHERICAL-FUNCTIONS ON SU(1,1)

Denote by L1(K\G/K) the algebra of spherical integrable functions on SU(1,1), with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in L1(K\G/K). In particular, we are interested in conditions on an ideal that ensure that it is all of L1(K\G/K), or that it is L0(1)(K\G\K). Spherical functions on SU(1,1) are naturally represented as radial functions on the unit disk D in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on D.