  The main goal is to interpret the Askey-Wilson function and the corresponding
transform pair on the quantum SU(1,1) group. A weight on the C^*-algebra of
continuous functions vanishing at infinity on the quantum SU(1,1) group is
studied, which is left and right invariant in a weak sense with respect to a
product defined using Wall functions. The Haar weight restricted to certain
subalgebras are explicitly determined in terms of an infinitely supported
Jackson integral and in terms of an infinitely supported Askey-Wilson type
measure. For the evaluation the spectral analysis of explicit unbounded doubly
infinite Jacobi matrices and some new summation formulas for basic
hypergeometric series are needed. The spherical functions are calculated in
terms of Askey-Wilson functions and big q-Jacobi functions. The corresponding
spherical Fourier transforms are identified with special cases of the big
q-Jacobi function transform and of the Askey-Wilson function transform.
