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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.
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