The problem of limited angle tomography in which a complete sinogram is not available is considered. This situation arises in many practical applications where tomographic projection over 180 deg is either physically unrealizable or infeasible. When a complete sinogram is not available, it is well known that the reconstructed images using common reconstruction algorithms, such as convolution back-projection (CBP), will have severe streak artifacts. We present a linear artificial neural network to extrapolate the missing part of the sinogram. Once the complete sinogram is obtained via extrapolation, standard reconstruction techniques such as CBP can be used to generate artifact-free reconstructions. The parameters of the neural network are designed using the sampling theory of signals with noncompact spectral support, the knowledge that complete sinograms have bow-tie-shaped spectral support and regularization. It is found that once designed, these parameters are data independent, especially for images of similar nature. For a sinogram with 2N angular views, each having M raysum per view, if 2L views are available, the computational requirement of the neural network is 4MNL only. The proposed neural network is much more efficient than other iterative algorithms, such as the method of projection onto convex sets; the Papoulis-Gerchberg algorithm; and the Clark-Palmer-Lawrence interpolation method; which requires computations of the order of kMNL, where k is the number of iterations.