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Draft:Slepian function

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Slepian functions are a class of spatio-spectrally concentrated functions (that is, space- or time-concentrated while spectrally bandlimited, or spectral-band-concentrated while space- or time-limited) 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 apodization 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 scalar, vector, and tensor forms.

Without reference to any of these particularities, let be a square-integrable function of physical space, and let represent Fourier transformation, such that and . Let the operators and project onto the space of spacelimited functions, , and the space of bandlimited functions, , respectively, whereby is an arbitrary nontrivial subregion of all of physical space, and am arbitrary nontrivial subregion of spectral space. Thus, the operator acts to spacelimit, and the operator acts to bandlimit the function .

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

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

and

given that and are each others' adjoints, and that and 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 , concentrated to the spatial domain within R, or spacelimited functions of the form , concentrated to the spectral domain within L.

Scalar Slepian functions in one dimension

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Let and its Fourier transform be strictly bandlimited in angular frequency between . Attempting to concentrate in the time domain, to be contained within the interval , amounts to maximizing

which is equivalent to solving either, in the frequency domain, the convolutional integral eigenvalue (Fredholm) equation

,

or the time- or space-domain version

.

Either of these can be transformed and rescaled to the dimensionless

.

The trace of the positive definite kernel is the sum of the infinite number of real and positive eigenvalues,

that is, the area of the concentration domain in time-frequency space.

Scalar Slepian functions in two Cartesian dimensions

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We use and its Fourier transform to denote a function that is strictly bandlimited to an arbitrary subregion of the spectral space of spatial wave vectors. Seeking to concentrate into a finite spatial region , of area , we must find the unknown functions for which

Maximize this Rayleigh quotient requires solving the Fredholm integral equation

The corresponding problem in the spatial domain is


Scalar Slepian functions on the surface of a sphere

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Vectorial and tensorial Slepian functions

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References

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

V. Michel. Spherical Slepian Functions, in Lectures on Constructive Approximation. Birkhäuser, 2012, doi:10.1007/978-0-8176-8403-7_8

C. T. Mullis and L. L. Scharf. Quadratic estimators of the power spec trum, in Advances in Spectrum Analysis and Array Processing, Vol. 1, chap. 1, pp. 1–57, ed. S. Haykin. Prentice-Hall, 1991, ISBN 978-0130074447

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes. The Art of Scientific Computing (Third Edition). Cambridge, 2007, ISBN 978-0-521-88068-8

F. J. Simons. Slepian functions and their use in signal estimation and spectral analysis. Handbook of Geomathematics, 2010, doi:10.1007/978-3-642-01546-5_30

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.

F. J. Simons and A. Plattner. Scalar and vector Slepian functions, spherical signal estimation and spectral analysis. Handbook of Geomathematics, 2015, doi:10.1007/978-3-642-54551-1_30