PHASE-SPACE LOCALIZATION AND DISCRETE REPRESENTATIONS OF WAVE-FIELDS

The high-frequency behavior of wave fields in free space is characterized by localization in phase space, exemplified by the description of geometrical optics in which distinct wave vectors are associated with the rays passing through each spatial point. I explore the manifestation of phase-space localization in discrete representations of the wave field, in particular the discrete Fourier transform (DFT) and the Gabor representation. A number of auxiliary concepts, such as spectral truncation, the Lagrange manifold, and the Landau-Pollak (LP) theorem, are described and exploited in the process of understanding the behavior of high-frequency fields, and it is shown that the Lagrange manifold is the source in phase space of the dominant contributions to these discrete representations in a number of specially selected examples. The LP theorem specifies the number of discrete degrees of freedom required for a given field to be approximated to a prescribed accuracy, and the theorem controls both the number of samples for DFT implementation and the number of Gabor coefficients required. The behavior of the Gabor coefficients away from the Lagrange manifold is studied for a Fresnel wave (quadratic phase variation on a line), and it is shown that these coefficients decay exponentially, signifying the localization of the Gabor representation. The number of operations required for computing the Gabor representation is compared with the fast-Fourier-transform (FFT) implementation of the DFT, with the LP dimension used as the common cardinality of the two representations, the result being that the FFT is asymptotically more efficient than the Gabor representation even after one allows for the localization of the latter. However; the Gabor representation can be used in circumstances in which the FFT is inapplicable, when the explicit localization is a significant advantage.