A weighted orthogonal system on the half line based on the Chebyshev
 rational functions is introduced. Basic results on Chebyshev rational
 approximations of several orthogonal projections and interpolations
 are established. To illustrate the potential of the Chebyshev
 rational spectral method, a model problem is considered both
 theoretically and numerically: error estimates for the Chebyshev
 rational spectral and pseudospectral methods are established;
 preliminary numerical results agree well with the theoretical
 estimates and demonstrate the effectiveness of this approach.