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This paper considers the problem to reconstruct and approximate multidimensional signals from nonuniformly distributed samples. Using multivariable spectral decompositions of functions in terms of empirical orthonormal basis functions we establish the exact recovery of a signal from its samples provided that the signal is band-limited in a well defined generic sense. The relation to sampling and approximate reconstruction of tensors is indicated. For non-band-limited signals expressions for the alias error are derived. An operator is introduced that reflects the alias sensitivity. The maximum alias sensitivity is characterized as the maximum eigenvalue of a suitably defined tensor operator. Results are illustrated by an example of signal reconstructions from partial measurements of a heat diffusion process.