  Wiener used the Poisson kernel for the Hermite polynomials to deal with the
classical Fourier transform. Askey, Atakishiyev and Suslov used this approach
to obtain a q-Fourier transform by using the continuous q-Hermite polynomials.
Rahman and Suslov extended this result by taking the Askey--Wilson polynomials,
considered to be the most general continuous classical orthogonal polynomials.
The theory of q-Fourier transformation is further extended here by considering
a nonsymmetric version of the Poisson kernel with Askey--Wilson polynomials.
This approach enables us to obtain some new results, for example, the complex
and real orthogonalities of these kernels.
