Wigner Measures in the Discrete Setting: High-Frequency Analysis of Sampling and Reconstruction Operators

The goal of this article is that of understanding how the oscillation and concentration effects developed by a sequence of functions in Rd are modified by the action of Sampling and Reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high frequency behavior of bounded sequences in L2 ( Rd ) . We actually present explicit formulas that make possible to compute such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.