Spatiospectral concentration in the Cartesian plane

We pose and solve the analogue of Slepian’s time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.

💡 Research Summary

The paper extends David Slepian’s classic time‑frequency concentration problem from one dimension to the full two‑dimensional Cartesian plane, providing a rigorous mathematical framework and practical algorithms for constructing functions that are simultaneously limited in both space and spectrum. The authors begin by defining two arbitrary closed sets: a spatial region (R\subset\mathbb{R}^2) where the energy of a function is to be concentrated, and a spectral region (\Omega\subset\mathbb{R}^2) that bounds the support of its Fourier transform. For a strictly band‑limited function (f(\mathbf{x})) (i.e., (\hat f(\mathbf{k})=0) for (\mathbf{k}\notin\Omega)) they introduce the concentration ratio

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