Approximations using the generalized Laguerre polynomials are
investigated in this paper. Error estimates for various orthogonal
projections are established. These estimates generalize and
improve previously published results on the Laguerre
approximations. As an example of applications, a mixed
Laguerre-Fourier spectral method for the Helmholtz equation in an
exterior domain is analyzed and implemented. The proposed method
enjoys  optimal error estimates, and with suitable basis
functions, leads to a sparse and symmetric linear system.