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Slepian functions are a class of spatio-spectrally concentrated functions that form an orthogonal basis for bandlimited or spacelimited spaces. They are widely used as basis functions for approximation and in linear inverse problems, and as tapers or window functions in quadratic problems of spectral density estimation.

Slepian functions exist in discrete and continuous varieties, in one, two, and three dimensions, in Cartesian and spherical geometry, on surfaces and in volumes, and in vector and tensor forms. Without reference to these particularities, let ${\displaystyle f}$ be a square-integrable function of physical space, and let ${\displaystyle {\mathcal {H}}}$ represent Fourier transformation, such that ${\displaystyle {\hat {f}}={\mathcal {H}}f}$ and ${\displaystyle {\mathcal {H}}^{-1}{\hat {f}}=f}$. Let the operators ${\displaystyle {\mathcal {R}}}$ and ${\displaystyle {\mathcal {L}}}$, project onto the space of spacelimited functions, ${\displaystyle {\mathcal {S}}_{R}}$, and the space of bandlimited functions, ${\displaystyle {\mathcal {S}}_{L}}$, respectively, whereby ${\displaystyle R}$ is an arbitrary nontrivial subregion of all of physical space, and ${\displaystyle L}$ a subregion of spectral space. Thus, the operator ${\displaystyle {\mathcal {R}}}$ acts to spacelimit, and the operator ${\displaystyle {\mathcal {H}}^{-1}{\mathcal {L}}{\mathcal {H}}}$ acts to bandlimit the function ${\displaystyle f}$.

Slepian's quadratic spectral concentration problem aims to maximize the concentration of spectral power to a target region ${\displaystyle L}$ for a function that is spatially limited to a target region ${\displaystyle R}$. Conversely, Slepian's spatial concentration problem maximizes the spatial concentration to ${\displaystyle R}$ of a function bandlimited to ${\displaystyle L}$. Using ${\displaystyle \langle \cdot ,\cdot \rangle }$ for the inner product both in the space and the spectral domain, both problems are stated equivalently as

${\displaystyle \lambda ={\frac {\langle {\mathcal {R}}{\mathcal {H}}^{-1}{\mathcal {L}}\,{\hat {f}},{\mathcal {R}}{\mathcal {H}}^{-1}{\mathcal {L}}\,{\hat {f}}\,\rangle }{\langle {\mathcal {H}}^{-1}{\mathcal {L}}\,{\hat {f}},{\mathcal {H}}^{-1}{\mathcal {L}}\,{\hat {f}}\,\rangle }}={\frac {\langle {\mathcal {L}}{\mathcal {H}}{\mathcal {R}}f,{\mathcal {L}}{\mathcal {H}}{\mathcal {R}}f\rangle }{\langle {\mathcal {H}}{\mathcal {R}}f,{\mathcal {H}}{\mathcal {R}}f\rangle }}={\mbox{maximum}}.}$

The equivalent spectral-domain and spatial-domain eigenvalue equations are

${\displaystyle ({\mathcal {L}}{\mathcal {H}}{\mathcal {R}}{\mathcal {H}}^{-1\!}{\mathcal {L}})({\mathcal {L}}\,{\hat {f}}\,)=\lambda ({\mathcal {L}}\,{\hat {f}}\,)}$ and ${\displaystyle ({\mathcal {R}}{\mathcal {H}}^{-1\!}{\mathcal {L}}{\mathcal {H}}{\mathcal {R}})({\mathcal {R}}f)=\lambda ({\mathcal {R}}f),}$

given that ${\displaystyle {\mathcal {H}}}$ and ${\displaystyle {\mathcal {H}}^{-1}}$ are each others' adjoints, and that ${\displaystyle {\mathcal {R}}}$ and ${\displaystyle {\mathcal {L}}}$ are self-adjoint and idempotent.

The Slepian functions are solutions to either of these types of equations with positive-definite kernels, that is, they are bandlimited functions ${\displaystyle {\hat {g}}={\mathcal {L}}{\hat {f}}}$ or spacelimited functions of the form ${\displaystyle h={\mathcal {R}}f}$.

Scalar Slepian functions in one dimension

Scalar Slepian functions in two Cartesian dimensions

Scalar Slepian functions on the surface of a sphere

Vectorial and tensorial Slepian functions

References

I. Daubechies. Ten Lectures on Wavelets. SIAM, 1992, ISBN  0-89871-274-2.

F. J. Simons, M. A. Wieczorek and F. A. Dahlen. Spatiospectral concentration on a sphere. SIAM Review, 2006, doi: 10.1137/S0036144504445765.

F. J. Simons and D. V. Wang. Spatiospectral concentration in the Cartesian plane. Int. J. Geomath, 2011, doi: 10.1007/s13137-011-0016-z.