Discrete fourier transform tensors and their ranks

We introduce a tensor generalization of the matrix discrete Fourier transform (DFT) which we call the collapsed DFT (CDFT) tensor. The CDFT tensor is different from the standard even order DFT tensor (except when the order is two). We study the action and compute the rank of the DFT tensor. Our main results for the CDFT tensors include their action, monotonicity of rank with respect to order, bounds on rank, and exact computation of rank of third order CDFT tensors when the dimension of the underlying space is a prime number.