We study an extension of the classical Paley-Wiener space structure, which is
based on bilinear expansions of integral kernels into biorthogonal sequences of
functions. The structure includes both sampling expansions and Fourier-Neumann
type series as special cases. Concerning applications, several new results are
obtained. From the Dunkl analogue of Gegenbauer's expansion of the plane wave,
we derive sampling and Fourier-Neumann type expansions and an explicit closed
formula for the spectrum of a right inverse of the Dunkl operator. This is done
by stating the problem in such a way it is possible to use the technique due to
Ismail and Zhang. Moreover, we provide a $q$-analogue of the Fourier-Neumann
expansions in $q$-Bessel functions of the third type. In particular, we obtain
a $q$-linear analogue of Gegenbauer's expansion of the plane wave by using
$q$-Gegenbauer polynomials defined in terms of little $q$-Jacobi polynomials.