On a general q-Fourier transformation with nonsymmetric kernels
  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. 
Richard A. Askey, Mizan Rahman, Serge\u{\i} K. Suslov