Abstract
The classical sampling theorems state that bandlimited square integrable functions of one or two variables can be reconstructed from their samples taken at suitably dense equally spaced sets of nodes. In one dimension, the uniform sampling points can be replaced by certain irregularly spaced ones. By making use of the properties of bandlimited functions and a generalization of an inequality of Nikol'skii shown in [22], in this part a nonuniform sampling theorem for two-dimensional square integrable functions bandlimited to a parallelogram centered at (0,0) is established. The reconstruction formula is obtained by iterative application of sampling formulae for functions of one variable. It is shown that the resulting series converges absolutely provided the sampling points are not too wildly scattered. Several applications are worked out, including those for hexagonal and octogonal sets of sampling points. Further generalizations are discussed
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