Fourier Transforms on the Quantum $\text{SU}(1,1)$ Group

Abstract

The main goal is to interpret the Askey–Wilson function and the corresponding transform pair on the quantum $\text{SU}(1,1)$ group. A weight on the ${\mathrm{C}}^{\ast }$-algebra of continuous functions vanishing at infinity on the quantum $\text{SU}(1,1)$ 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.