  In this article, we obtain a complete list of inequivalent irreducible
representations of the compact quantum group $U_q(2)$ for non-zero complex
deformation parameters $q$, which are not roots of unity. The matrix
coefficients of these representations are described in terms of the little
$q$-Jacobi polynomials. The Haar state is shown to be faithful and an
orthonormal basis of $L^2(U_q(2))$ is obtained. Thus, we have an explicit
description of the Peter-Weyl decomposition of $U_q(2)$. As an application, we
discuss the Fourier transform and establish the Plancherel formula. We also
describe the decomposition of the tensor product of two irreducible
representations into irreducible components. Finally, we classify the compact
quantum groups $U_q(2)$.
